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800 |
A cohomology for vector valued differential forms
|
math.DG
|
A rather simple natural outer derivation of the graded Lie algebra of all
vector valued differential forms with the Fr\"olicher-Nijenhuis bracket turns
out to be a differential and gives rise to a cohomology of the manifold, which
is functorial under local diffeomorphisms. This cohomology is determined as the
direct product of the de Rham cohomology space and the graded Lie algebra of
"traceless" vector valued differential forms, equipped with a new natural
differential concomitant as graded Lie bracket. We find two graded Lie algebra
structures on the space of differential forms. Some consequences and related
results are also discussed.
|
math
|
801 |
Nonunique tangent maps at isolated singularities of harmonic maps
|
math.DG
|
Shoen and Uhlenbeck showed that ``tangent maps'' can be defined at singular
points of energy minimizing maps. Unfortunately these are not unique, even for
generic boundary conditions. Examples are discussed which have isolated
singularities with a continuum of distinct tangent maps.
|
math
|
802 |
Graded derivations of the algebra of differential forms associated with a connection
|
math.DG
|
In the main part of this paper a connection is just a fiber projection onto a
(not necessarily integrable) distribution or sub vector bundle of the tangent
bundle. Here curvature is computed via the Froelicher-Nijenhuis bracket, and it
is complemented by cocurvature and the Bianchi identity still holds. In this
situation we determine the graded Lie algebra of all graded derivations over
the horizontal projection of a connection and we determine their commutation
relations. Finally, for a principal connection on a principal bundle and the
induced connection on an associated bundle we show how one may pass from one to
the other. The final results relate derivations on vector bundle valued forms
and derivations over the horizontal projection of the algebra of forms on the
principal bundle with values in the standard vector space.
|
math
|
803 |
The action of the diffeomorphism group on the space of immersions
|
math.DG
|
We study the action of the diffeomorphism group $\Diff(M)$ on the space of
proper immersions $\Imm_{\text{prop}}(M,N)$ by composition from the right. We
show that smooth transversal slices exist through each orbit, that the quotient
space is Hausdorff and is stratified into smooth manifolds, one for each
conjugacy class of isotropy groups.
|
math
|
804 |
The relation between systems and associated bundles
|
math.DG
|
It is shown that a strong system of vector fields on a fiber bundle in the
sense of [Modugno, M. Systems of connections and invariant lagrangians. In:
Differential geometric methods in theoretical physics, Proc. 15th Int. Conf.,
DGM, Clausthal/FRG 1986, 518-534 World Scientific Publishing Co. (1987)] is
induced from a principal fiber bundle if and only if each vertical vector field
of the system is complete.
|
math
|
805 |
Commutators of flows and fields
|
math.DG
|
The well known formula $[X,Y]=\tfrac12\tfrac{\partial^2}{\partial t^2}|_0
(\Fl^Y_{-t}\o\Fl^X_{-t}\o\Fl^Y_t\o\Fl^X_t)$ for vector fields $X$, $Y$ is
generalized to arbitrary bracket expressions and arbitrary curves of local
diffeomorphisms.
|
math
|
806 |
Geodesics on spaces of almost hermitian structures
|
math.DG
|
A natural metric on the space of all almost hermitian structures on a given
manifold is investigated.
|
math
|
807 |
One cannot hear the shape of a drum
|
math.DG
|
We use an extension of Sunada's theorem to construct a nonisometric pair of
isospectral simply connected domains in the Euclidean plane, thus answering
negatively Kac's question, ``can one hear the shape of a drum?'' In order to
construct simply connected examples, we exploit the observation that an
orbifold whose underlying space is a simply connected manifold with boundary
need not be simply connected as an orbifold.
|
math
|
808 |
Characteristic classes for $G$-structures
|
math.DG
|
Let $G\subset GL(V)$ be a linear Lie group with Lie algebra $\frak g$ and let
$A(\frak g)^G$ be the subalgebra of $G$-invariant elements of the associative
supercommutative algebra $A(\frak g)= S(\frak g^*)\otimes \La(V^*)$. To any
$G$-structure $\pi:P\to M$ with a connection $\omega$ we associate a
homomorphism $\mu_\omega:A(\frak g)^G\to \Omega(M)$. The differential forms
$\mu_\omega(f)$ for $f\in A(\frak g)^G$ which are associated to the
$G$-structure $\pi$ can be used to construct Lagrangians. If $\omega$ has no
torsion the differential forms $\mu_\omega(f)$ are closed and define
characteristic classes of a $G$-structure. The induced homomorphism
$\mu'_\omega:A(\g)^G\to H^*(M)$ does not depend on the choice of the
torsionfree connection $\omega$ and it is the natural generalization of the
Chern Weil homomorphism.
|
math
|
809 |
Adding handles to the helicoid
|
math.DG
|
There exist two new embedded minimal surfaces, asymptotic to the helicoid.
One is periodic, with quotient (by orientation-preserving translations) of
genus one. The other is nonperiodic of genus one.
|
math
|
810 |
Differential geometry of $\frak g$-manifolds
|
math.DG
|
An action of a Lie algebra $\frak g$ on a manifold $M$ is just a Lie algebra
homomorphism $\zeta:\frak g\to \frak X(M)$. We define orbits for such an
action. In general the space of orbits $M/\frak g$ is not a manifold and even
has a bad topology. Nevertheless for a $\frak g$-manifold with equidimensional
orbits we treat such notions as connection, curvature, covariant
differentiation, Bianchi identity, parallel transport, basic differential
forms, basic cohomology, and characteristic classes, which generalize the
corresponding notions for principal $G$-bundles. As one of the applications, we
derive a sufficient condition for the projection $M\to M/\frak g$ to be a
bundle associated to a principal bundle.
|
math
|
811 |
A theory of characteristic currents associated with a singular connection
|
math.DG
|
This note announces a general construction of characteristic currents for
singular connections on a vector bundle. It develops, in particular, a
Chern-Weil-Simons theory for smooth bundle maps $\alpha : E \rightarrow F$
which, for smooth connections on $E$ and $F$, establishes formulas of the type
$$ \phi \ = \ \text{\rm Res}_{\phi}\Sigma_{\alpha} + dT. $$ Here $\phi$ is a
standard charactersitic form, $\text{Res}_{\phi}$ is an associated smooth
``residue'' form computed canonically in terms of curvature, $\Sigma_{\alpha}$
is a rectifiable current depending only on the singular structure of $\alpha$,
and $T$ is a canonical, functorial transgression form with coefficients in
$\loc$. The theory encompasses such classical topics as: Poincar\'e-Lelong
Theory, Bott-Chern Theory, Chern-Weil Theory, and formulas of Hopf.
Applications include:\ \ a new proof of the Riemann-Roch Theorem for vector
bundles over algebraic curves, a $C^{\infty}$-generalization of the
Poincar\'e-Lelong Formula, universal formulas for the Thom class as an
equivariant characteristic form (i.e., canonical formulas for a de Rham
representative of the Thom class of a bundle with connection), and a
Differentiable Riemann-Roch-Grothendieck Theorem at the level of forms and
currents. A variety of formulas relating geometry and characteristic classes
are deduced as direct consequences of the theory.
|
math
|
812 |
On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces
|
math.DG
|
We develop an asymptotic expansion of the spectral measures on a degenerating
family of hyperbolic Riemann surfaces of finite volume. As an application of
our results, we study the asymptotic behavior of weighted counting functions,
which, if $M$ is compact, is defined for $w \geq 0$ and $T > 0$ by $$N_{M,w}(T)
= \sum\limits_{\lambda_n \leq T}(T-\lambda_n)^w $$ where $\{\lambda_n\}$ is the
set of eigenvalues of the Laplacian which acts on the space of smooth functions
on $M$. If $M$ is non-compact, then the weighted counting function is defined
via the inverse Laplace transform. Now let $M_{\ell}$ denote a degenerating
family of compact or non-compact hyperbolic Riemann surfaces of finite volume
which converges to the non-compact hyperbolic surface $M_{0}$. As an example of
our results, we have the following theorem: There is an explicitly defined
function $G_{\ell,w}(T)$ which depends solely on $\ell$, $w$, and $T$ such that
for $w > 3/2$ and $T>0$, we have $$N_{M_{\ell},w}(T) = G_{\ell,w}(T)
+N_{M_{0},w}(T) +o(1)$$ for $\ell \to 0$. We also consider the setting when $w
< 3/2$, and we obtain a new proof of the continuity of small eigenvalues on
degenerating hyperbolic Riemann surfaces of finite volume.
|
math
|
813 |
Differential geometry of Cartan connections
|
math.DG
|
For a more general notion of Cartan connection we define characteristic
classes, we investigate their relation to usual characteristic classes.
|
math
|
814 |
A New Construction of Isospectral Riemannian Nilmanifolds with Examples
|
math.DG
|
We present a new construction for obtaining pairs of higher-step isospectral
Riemannian nilmanifolds and compare several resulting new examples. In
particular, we present new examples of manifolds that are isospectral on
functions, but not isospectral on one-forms.
|
math
|
815 |
Geometric Zeta Functions, $L^2$-Theory, and Compact Shimura Manifolds
|
math.DG
|
We define geometric zeta functions for locally symmetric spaces as
generalizations of the zeta functions of Ruelle and Selberg. As a special value
at zero we obtain the Reidemeister torsion of the manifold. For hermitian
spaces these zeta functions have as special value the quotient of the
holomorphic torsion of Ray and Singer and the holomorphic $L^2$-torsion, where
the latter is defined via the $L^2$-theory of Atiyah. For higher fundamental
rank twisted torsion numbers appear.
|
math
|
816 |
Geodesic Conjugacy in two-step nilmanifolds
|
math.DG
|
Two Riemannian manifolds are said to have $C^k$-conjugate geodesic flows if
there exist an $C^k$ diffeomorphism between their unit tangent bundles which
intertwines the geodesic flows. We obtain a number of rigidity results for the
geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step
nilmanifolds the geodesic flow is $C^2$ rigid. For special classes of 2-step
nilmanifolds, we show that the geodesic flow is $C^0$ or $C^2$ rigid. In
particular, there exist continuous families of 2-step nilmanifolds whose
Laplacians are isospectral but whose geodesic flows are not $C^0$ conjugate.
|
math
|
817 |
Filling-invariants at infinity for manifolds of nonpositive curvature
|
math.DG
|
In this paper we construct and study isoperimetric functions at infinity for
Hadamard manifolds. These quasi-isometry invariants give a measure of the
spread of geodesics in such a manifold.
|
math
|
818 |
Arithmeticity, Discreteness and Volume
|
math.DG
|
We give an arithmetic criterion which is sufficient to imply the discreteness
of various two-generator subgroups of $PSL(2,{\bold C})$. We then examine
certain two-generator groups which arise as extremals in various geometric
problems in the theory of Kleinian groups, in particular those encountered in
efforts to determine the smallest co-volume, the Margulis constant and the
minimal distance between elliptic axes. We establish the discreteness and
arithmeticity of a number of these extremal groups, the associated minimal
volume arithmetic group in the commensurability class and we study whether or
not the axis of a generator is simple.
|
math
|
819 |
On the Margulis constant for Kleinian groups, I curvature
|
math.DG
|
The Margulis constant for Kleinian groups is the smallest constant $c$ such
that for each discrete group $G$ and each point $x$ in the upper half space
${\bold H}^3$, the group generated by the elements in $G$ which move $x$ less
than distance c is elementary. We take a first step towards determining this
constant by proving that if $\langle f,g \rangle$ is nonelementary and discrete
with $f$ parabolic or elliptic of order $n \geq 3$, then every point $x$ in
${\bold H}^3$ is moved at least distance $c$ by $f$ or $g$ where
$c=.1829\ldots$. This bound is sharp.
|
math
|
820 |
A note on Carnot geodesics in nilpotent Lie groups
|
math.DG
|
We show that strictly abnormal geodesics arise in graded nilpotent Lie
groups. We construct such a group, for which some Carnot geodesics are strictly
abnormal; in fact, they are not normal in any subgroup. In the step-2 case we
also prove that these geodesics are always smooth. Our main technique is based
on the equations for the normal and abnormal curves, that we derive (for any
Lie group) explicitly in terms of the structure constants.
|
math
|
821 |
Complete embedded minimal surfaces of finite total curvature
|
math.DG
|
We survey what is known about minimal surfaces in $\bold R^3 $ that are
complete, embedded, and have finite total curvature. The only classically known
examples of such surfaces were the plane and the catenoid. The discovery by
Costa, early in the last decade, of a new example that proved to be embedded
sparked a great deal of research in this area. Many new examples have been
found, even families of them, as will be described below. The central question
has been transformed from whether or not there are any examples except surfaces
of rotation to one of understanding the structure of the space of examples.
|
math
|
822 |
Grafting, harmonic maps, and projective structures on surfaces
|
math.DG
|
Grafting is a surgery on Riemann surfaces introduced by Thurston which
connects hyperbolic geometry and the theory of projective structures on
surfaces. We will discuss the space of projective structures in terms of the
Thurston's geometric parametrization given by grafting. From this approach we
will prove that on any compact Riemann surface with genus greater than $1$
there exist infinitely many projective structures with Fuchsian holonomy
representations. In course of the proof it will turn out that grafting is
closely related to harmonic maps between surfaces.
|
math
|
823 |
The Marked Length Spectrum Versus the Laplace Spectrum on Forms on Riemannian Nilmanifolds
|
math.DG
|
The subject of this paper is the relationship among the marked length
spectrum, the length spectrum, the Laplace spectrum on functions, and the
Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show
that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in
this class has the same marked length spectrum, they necessarily share the same
Laplace spectrum on functions. In contrast, we present the first example of a
pair of isospectral Riemannian manifolds with the same marked length spectrum
but not the same spectrum on one-forms. Outside of the standard spheres vs. the
Zoll spheres, which are not even isospectral, this is the only example of a
pair of Riemannian manifolds with the same marked length spectrum, but not the
same spectrum on forms. This partially extends and partially contrasts the work
of Eberlein, who showed that on two-step nilmanifolds, the same marked length
spectrum implies the same Laplace spectrum both on functions and on forms.
|
math
|
824 |
Hodge theory in the Sobolev topology for the de Rham complex
|
math.DG
|
The authors study the Hodge theory of the exterior differential operator $d$
acting on $q$-forms on a smoothly bounded domain in $\RR^{N+1}$, and on the
half space $\rnp$. The novelty is that the topology used is not an $L^2$
topology but a Sobolev topology. This strikingly alters the problem as compared
to the classical setup. It gives rise to a boundary-value problem belonging to
a class of problems first introduced by Vi\v{s}ik and Eskin, and by Boutet de
Monvel.
|
math
|
825 |
Homogeneous Special Geometry
|
math.DG
|
Motivated by the physical concept of special geometry two mathematical
constructions are studied, which relate real hypersurfaces to tube domains and
complex Lagrangean cones respectively. Me\-thods are developed for the
classification of homogeneous Riemannian hypersurfaces and for the
classification of linear transitive reductive algebraic group actions on pseudo
Riemannian hypersurfaces. The theory is applied to the case of cubic
hypersurfaces, which is the case most relevant to special geometry, obtaining
the solution of the two classification problems and the description of the
corresponding homogeneous special K\"ahler manifolds.
|
math
|
826 |
Quasiconformality and geometrical finiteness in Carnot--Carathéodory and negatively curved spaces
|
math.DG
|
The paper sketches a recent progress and formulates several open problems in
studying equivariant quasiconformal and quasisymmetric homeomorphisms in
negatively curved spaces as well as geometry and topology of noncompact
geometrically finite negatively curved manifolds and their boundaries at
infinity having Carnot--Carath\'eodory structures. Especially, the most
interesting are complex hyperbolic manifolds with Cauchy--Riemannian structure
at infinity, which occupy a distinguished niche and whose properties make them
surprisingly different from real hyperbolic ones.
|
math
|
827 |
An estimate for the Gauss curvature of minimal surfaces in $\mathbb R^m$ whose Gauss map omits a set of hyperplanes
|
math.DG
|
We give an estimate of the Gauss curvature for minimal surfaces in ${\mathbb
R}^m$ whose Gauss map omits more than $m(m+1)/2$ hyperplanes in ${\mathbb
P}^{m-1}({\mathbb C})$.
|
math
|
828 |
The Singly Periodic Genus-One Helicoid
|
math.DG
|
We prove the existence of a complete, embedded, singly periodic minimal
surface, whose quotient by vertical translations has genus one and two ends.
The existence of this surface was announced in our paper in {\it Bulletin of
the AMS}, 29(1):77--84, 1993. Its ends in the quotient are asymptotic to one
full turn of the helicoid, and, like the helicoid, it contains a vertical line.
Modulo vertical translations, it has two parallel horizontal lines crossing the
vertical axis. The nontrivial symmetries of the surface, modulo vertical
translations, consist of: $180^\circ$ rotation about the vertical line;
$180^\circ$ rotation about the horizontal lines (the same symmetry); and their
composition.
|
math
|
829 |
Asymptotic geometry and conformal types of Carnot--Carathéodory spaces
|
math.DG
|
An intrinsic definition in terms of conformal capacity is proposed for the
conformal type of a Carnot--Carath\'eodory space (parabolic or hyperbolic).
Geometric criteria of conformal type are presented. They are closely related to
the asymptotic geometry of the space at infinity and expressed in terms of the
isoperimetric function and the growth of the area of geodesic spheres. In
particular, it is proved that a sub-Riemannian manifold admits a conformal
change of metric that makes it into a complete manifold of finite volume if and
only if the manifold is of conformally parabolic type. Further applications are
discussed, such as the relation between local and global invertibility
properties of quasiconformal immersions (the global homeomorphism theorem).
|
math
|
830 |
Mixing Mathematics and Materials
|
math.DG
|
Recent uses of differential geometry in materials science are reviewed here,
in particular the September issue of the Phil. Trans. Royal Soc., entitled
``Curvature and chemical Structure.''
|
math
|
831 |
The Jacobi flow
|
math.DG
|
The geodesic flow on the tangent bundle is the flow of a certain vector field
which is called the spray $S:TM\to TTM$. The flow lines of the vector field
$\ka_{TM}\o TS:TTM\to TTTM$ project to the Jacobi fields on $TM$. This could be
called the Jacobi flow.
|
math
|
832 |
Flattening and subanalytic sets in rigid analytic geometry
|
math.DG
|
Let K be an algebraically closed field endowed with a complete
non-archimedean norm with valuation ring R. Let f:Y -> X be a map of K-affinoid
varieties. In this paper we study the analytic structure of the image f(Y) in
X; such an image is a typical example of a subanalytic set. We show that the
subanalytic sets are precisely the D-semianalytic sets, where D is the
truncated division function first introduced by Denef and van den Dries. This
result is most conveniently stated as a Quantifier Elimination result for the
valuation ring R in an analytic expansion of the language of valued fields.
|
math
|
833 |
Verdier stratifications and [wf]-stratification in o-minimal structures
|
math.DG
|
We prove the existence of Verdier stratifications for sets definable in any
o-minimal structure on (R, +, .). It is also shown that the Verdier condition
(w) implies the Whitney condition (b) in o-minimal structures on (R, +, .). As
a consequence the Whitney Stratification Theorem holds. The existence of
(wf)-stratification of functions definable in polynomially bounded o-minimal
structures is presented.
|
math
|
834 |
On the bifurcation sets of functions definable in o-minimal structures
|
math.DG
|
Let g:X -> Y be a smooth (i.e. C^\infty differentiable) map between two
smooth manifolds. In analogy with the case of complex polynomial functions, we
say that y_0 in Y is a typical value of g if there exists an open neighbourhood
U of y_0 in Y, such that the restriction g:g^{-1}(U) -> U is a C^\infty trivial
fibration. If y_0 in Y is not a typical value of g, then y_0 is called an
atypical value of g. We denote by B_g the bifurcation set of g, i.e. the set of
atypical values of g. In the case of a complex polynomial function f:C^n -> C
it is known that B_f is a finite set. It was previously proved that the
bifurcation sets of real polynomial functions are also finite.
The aim of this note is to show that the bifurcation set B_f of a smooth
definable function f:R^n -> R is finite .
|
math
|
835 |
Closure of rigid semianalytic sets
|
math.DG
|
Let K be an algebraically closed field of characteristic zero, endowed with a
complete nonarchimedean norm. Let X be a K-rigid analytic variety and \Sigma a
semianalytic subset of X. Then the closure of \Sigma in X with respect to the
canonical topology is again semianalytic. The proof uses Embedded Resolution of
Singularities.
|
math
|
836 |
Motion by weighted mean curvature is affine invariant
|
math.DG
|
Suppose curves are moving by curvature in a plane, but one embeds the plane
in $R^3$ and looks at the plane from an angle. Then circles shrinking to a
round point would appear to be ellipses shrinking to an ``elliptical point,''
and the surface energy would appear to be anisotropic as would the mobility.
The result of this paper is that if one uses the apparent surface energy and
the apparent mobility, then the motion by weighted curvature with mobility in
the apparent plane is the same as motion by curvature in the original plane but
then viewed from the angle. This result applies not only to the isotropic case
but to arbitrary surface energy functions and mobilities in the plane, to
surfaces in 3-space, and (in the case that the surface energy function is twice
differentiable) to the case of motion viewed through distorted lenses (i.e.,
diffeomorphisms) as well. This result is to be contrasted with an earlier
result which states that for area-preserving affine transformations of the
plane where the energy and mobility are NOT also transformed, motion by
curvature to the power 1/3 (rather than 1) is invariant.
|
math
|
837 |
Vertex theorems for capillary drops on support planes
|
math.DG
|
We consider a capillary drop that contacts several planar bounding walls so
as to produce singularities (vertices) in the boundary of its free surface. It
is shown under various conditions that when the number of vertices is less than
or equal to three, then the free surface must be a portion of a sphere. These
results extend the classical theorem of H. Hopf on constant mean curvature
immersions of the sphere. The conclusion of sphericity cannot be extended to
more than three vertices, as we show by examples.
|
math
|
838 |
Ward's solitions
|
math.DG
|
Using the `Riemann Problem with zeros' method, Ward has constructed exact
solutions to a (2+1)-dimensional integrable Chiral Model, which exhibit
solitons with nontrivial scattering. We give a correspondence between what we
conjecture to be all pure soliton solutions and certain holomorphic vector
bundles on a compact surface.
|
math
|
839 |
Functions on space curves
|
math.DG
|
We classify simple singularities of functions on space curves. We show that
their bifurcation sets have properties very similar to those of functions on
smooth manifolds and complete intersections [1,2]: the k(pi, 1)-theorem for the
bifurcations diagram of functions is true, and both this diagram and the
discriminant are Saito's free divisors.
|
math
|
840 |
Embedded minimal ends asymptotic to the helicoid
|
math.DG
|
The ends of a complete embedded minimal surface of {\em finite total
curvature} are well understood (every such end is asymptotic to a catenoid or
to a plane). We give a similar characterization for a large class of ends of
{\em infinite total curvature}, showing that each such end is asymptotic to a
helicoid. The result applies, in particular, to the genus one helicoid and
implies that it is embedded outside of a compact set in ${\mathbb R}^3$.
|
math
|
841 |
Generalization of the Chekanov theorem: diameters of immersed manifolds and wave fronts
|
math.DG
|
The Chekanov theorem generalizes the classic Lyusternik-Shnirel'man and Morse
theorems concerning critical points of a smooth function on a closed manifold.
A Legendrian submanifold \Lambda of space of 1-jets of the functions on a
manifold M defines a multi-valued function whose graph is the projection of
\Lambda in J^0 M = M x R. The Chekanov theorem asserts that if \Lambda is
homotopic to the 1-jet of a smooth function in the class of embedded Legendrian
manifolds, then such a graph of a multi-valued function must have a lot of
points (their number is determined by the topology of M) at which the tangent
plane to the graph is parallel to M \times 0.
In the present paper a similar estimate is proved for a wider class of
Legendrian manifolds. We consider Legendrian manifolds homotopic (in the class
of embedded Legendrian manifolds) to Legendrian manifolds specified by
generating families.
|
math
|
842 |
Variational problems for Riemannian functionals and arithmetic groups
|
math.DG
|
In this paper we introduce a new approach to variational problems on the
space Riem(M^n) of Riemannian structures (i.e. isometry classes of Riemannan
metrics) on any fixed compact manifold M^n of dimension n >= 5. This approach
often enables one to replace the considered variational problem on Riem(M^n)
(or on some subset of Riem(M^n)) by the same problem but on spaces Riem(N^n)
for every manifold N^n from a class of compact manifolds of the same dimension
and with the same homology as M^n but with the following two useful properties:
(1) If \nu is any Riemannian structure on any manifold N^n from this class such
that Ric_(N^n,\nu) >= -(n-1), then the volume of (N^n,\nu) is greater than one;
and (2) Manifolds from this class do not admit Riemannian metrics of
non-negative scalar curvature.
As a first application we prove a theorem which can be informally explained
as follows: Let M be any compact connected smooth manifold of dimension greater
than four, M et(M) be the space of isometry classes of compact metric spaces
homeomorphic to M endowed with the Gromov-Hausdorff topology, Riem_1(M) in M
et(M ) be the space of Riemannian structures on M such that the absolute values
of sectional curvature do not exceed one, and R_1(M) denote the closure of
Riem_1(M) in M et(M ). Then diameter regarded as a functional on R_1(M) has
infinitely many "very deep" local minima.
|
math
|
843 |
Regular infinite dimensional Lie groups
|
math.DG
|
Regular Lie groups are infinite dimensional Lie groups with the property that
smooth curves in the Lie algebra integrate to smooth curves in the group in a
smooth way (an `evolution operator' exists). Up to now all known smooth Lie
groups are regular. We show in this paper that regular Lie groups allow to push
surprisingly far the geometry of principal bundles: parallel transport exists
and flat connections integrate to horizontal foliations as in finite
dimensions. As consequences we obtain that Lie algebra homomorphisms intergrate
to Lie group homomorphisms, if the source group is simply connected and the
image group is regular.
|
math
|
844 |
The Weierstrass representation of spheres in $R^3$, the Willmore numbers, and soliton spheres
|
math.DG
|
The Weierstrass representation for spheres in $\R^3$ and, in particular,
effective construction of immersions from data of spectral theory origin is
discussed. These data are related to Dirac operators on a plane and on an
infinite cylinder and these operators are just representations of Dirac
operators acting in spinor bundles over the two-sphere which is naturally
obtained as a completion of a plane or of a cylinder. Spheres described in
terms of Dirac operators with one-dimensional potentials on a cylinder are
completely studied and, in particular, for them a lower estimate of the
Willmore functional in terms of the dimension of the kernel of the
corresponding Dirac operator on a two-sphere is obtained. It is conjectured
that this estimate is valid for all Dirac operators on spheres and some
reasonings for this conjecture are discussed. In Appendix a criterion
distinguishing Weierstrass representations, of universal coverings of compact
surfaces of higher genera, converted into immersions of compact surfaces is
given.
|
math
|
845 |
No slices on the space of generalized connections
|
math.DG
|
On a fiber bundle without structure group the action of the gauge group (the
group of all fiber respecting diffeomorphisms) on the space of (generalized)
connections is shown not to admit slices.
|
math
|
846 |
Poisson structures on double Lie groups
|
math.DG
|
Lie bialgebra structures are reviewed and investigated in terms of the double
Lie algebra, of Manin- and Gau{\ss}-decompositions. The standard R-matrix in a
Manin decomposition then gives rise to several Poisson structures on the
correponding double group, which is investigated in great detail.
|
math
|
847 |
Lifting smooth curves over invariants for representations of compact Lie groups
|
math.DG
|
We show that one can lift locally real analytic curves from the orbit space
of a compact Lie group representation, and that one can lift smooth curves even
globally, but under an assumption.
|
math
|
848 |
A tangent bundle on diffeological spaces
|
math.DG
|
We define a subcategory of the category of diffeological spaces, which
contains smooth manifolds, the diffeomorphism subgroups and its coadjoint
orbits. In these spaces we construct a tangent bundle, vector fields and a de
Rham cohomology.
|
math
|
849 |
Hofer's diameter and Lagrangian intersections
|
math.DG
|
We prove that the group of Hamiltonian diffeomorphisms of the 2-sphere has
infinite diameter with respect to Hofer's metric. Our approach is based on the
theory of Lagrangian intersections.
|
math
|
850 |
Curvature of the Virasoro-Bott group
|
math.DG
|
We consider a natural Riemannian metric on the infinite dimensional manifold
of all embeddings from a manifold into a Riemannian manifold, and derive its
geodesic equation in the case $\Emb(\Bbb R,\Bbb R)$ which turns out to be
Burgers' equation. Then we derive the geodesic equation, the curvature, and the
Jacobi equation of a right invariant Riemannian metric on an infinite
dimensional Lie group, which we apply to $\Diff(\Bbb R)$, $\Diff(S^1)$, and the
Virasoro-Bott group. Many of these results are well known, the emphasis is on
conciseness and clarity.
|
math
|
851 |
On the cohomology of sl(m+1,R) acting on differential operators and sl(m+1,R)-equivariant symbol
|
math.DG
|
One computes the cohomology of the projective embedding of sl(m+1,R) acting
on the differential operators on densities on R^m of various weights. This
cohomology is non vanishing only for some special critical values of the
weights. This allows us first to explain some strange feature pointed out by
Gargoubi in his classification if the module of differential operators on the
line. This also allows us to provide a so called sl(m+1,R) invariant symbol for
differential operators acting on densities of non critical weights.
|
math
|
852 |
On the case of Goryachev-Chaplygin and new examples of integrable conservative systems on S^2
|
math.DG
|
The aim of this paper is to describe a class of conservative systems on $S^2$
possessing an integral cubic in momenta. We prove that this class of systems
consists off the case of Goryachev-Chaplygin, the one-parameter family of
systems which has been found by the author in the previous paper
(dg-ga/9711005) and a new two-parameter family of conservative systems on $S^2$
possessing an integral cubic in momenta.
|
math
|
853 |
Metrics of constant curvature 1 with three conical singularities on 2-sphere
|
math.DG
|
A necessary and sufficient condition for the existence and uniqueness of a
conformal metric on 2-sphere of constant curvature 1 and with three conical
singularities of prescribed order is given.
|
math
|
854 |
The outer derivation of a complex Poisson manifold
|
math.DG
|
We introduce a canonical outer vector field on a Poisson manifold, also due
independently to A. Weinstein. We view it as a global section of the sheaf of
Poisson vector fields modulo the subsheaf of hamiltonian vector fields. We
study this outer derivation mostly in the case of holomorphic Poisson
manifolds.
|
math
|
855 |
Cartan Spinor Bundles on Manifolds
|
math.DG
|
The aim of this paper is the construction of spinor bundles of Cartan type
over certain non-orientable manifolds.
|
math
|
856 |
Holomorphic spinors and the Dirac equation
|
math.DG
|
A closed spin K\"ahler manifold of positive scalar curvature with smallest
possible first eigenvalue of the Dirac operator is characterized by holomorphic
spinors. It is shown that on any spin K\"ahler-Einstein manifold each
holomorphic spinor is a finite sum of eigenspinors of the square of the Dirac
operator. Vanishing theorems for holomorphic spinors are proved.
|
math
|
857 |
Gluing theorems for anti-self-dual metrics
|
math.DG
|
In this paper we announce a gluing theorem for conformal structures with
anti-self-dual (ASD) Weyl tensor that applies in geometrical situations that
are more general than those considered by previous authors. By adapting a
method proposed by Floer, sufficient conditions are given for the existence of
ASD conformal structures on `generalized connected sums' of non-compact ASD
4-manifolds with cylindrical ends. The gluing theorem applies in particular to
give results about connected sums of ASD orbifolds along (isolated) singular
points.
|
math
|
858 |
A four dimensional example of Ricci-flat metric admitting almost-Kähler non-Káhler structure
|
math.DG
|
We construct an example of Ricci-flat almost-K\"ahler non-K\"ahler structure
in four dimensions.
|
math
|
859 |
Mathai-Quillen forms and Lefschetz theory
|
math.DG
|
Mathai-Quillen forms are used to give an integral formula for the Lefschetz
number of a smooth map of a closed manifold. Applied to the identity map, this
formula reduces to the Chern-Gauss-Bonnet theorem. The formula is computed
explicitly for constant curvature metrics. There is in fact a one-parameter
family of integral expressions. As the parameter goes to infinity, a
topological version of the heat equation proof of the Lefschetz fixed
submanifold formula is obtained. As the parameter goes to zero and under a
transversality assumption, a lower bound for the number of points mapped into
their cut locus is obtained. For diffeomorphisms with Lefschetz number unequal
to the Euler characteristic, this number is infinite for most metrics, in
particular for metrics of non-positive curvature.
|
math
|
860 |
The loop derivative as a curvature
|
math.DG
|
Recently, a set of tools has been developed with the purpose of the study of
Quantum Gravity. Until now, there have been very few attempts to put these
tools into a rigorous mathematical framework. This is the case, for example, of
the so called path bundle of a manifold. It is well known that this topological
principal bundle plays the role of a universal bundle for the reconstruction of
principal bundles and their connections. The path bundle is canonically endowed
with a parallel transport and associated with it important types of derivatives
have been considered by several authors: the Mandelstam derivative, the
connection derivative and the Loop derivative. In the present article we shall
give a unified viewpoint for all these derivatives by developing a
differentiable calculus on the path bundle. In particular we shall show that
the loop derivative is the curvature of a canonically defined one form that we
shall called the universal connection one form.
|
math
|
861 |
Higher analytic torsion of sphere bundles and continuous cohomology of $Diff(S^{2n-1})$
|
math.DG
|
Using the higher analytic torsion form of Bismut and Lott we construct a
characteristic class for smooth sphere bundles. We calculate this class in the
case where the sphere bundle comes from a complex vector bundle. Related to
these characteristic classes we define nontrivial continuous group cohomology
classes of the diffeomorphism group of odd dimensional spheres.
|
math
|
862 |
Symplectic reduction and a weighted multiplicity formula for twisted Spin$^c$-Dirac operators
|
math.DG
|
We extend our earlier work in [TZ1], where an analytic approach to the
Guillemin-Sternberg conjecture [GS] was developed, to cases where the
Spin$^c$-complex under consideration is allowed to be further twisted by
certain natural exterior power bundles. The main result is a weighted
quantization formula in the presence of commuting Hamiltonian actions. The
corresponding Morse type inequalities in holomorphic situations are also
established.
|
math
|
863 |
Singularities and bifurcations of 3-dimensional Poisson structure
|
math.DG
|
We give a normal form for families of 3-dimensional Poisson structures. This
allows us to classify singularities with nonzero 1-jet and typical
bifurcations. The Appendix contains corollaries on classification of families
of integrable 1-forms on $R^3
|
math
|
864 |
Orthogonal nets and Clifford algebras
|
math.DG
|
A Clifford algebra model for M"obius geometry is presented. The notion of
Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced,
and the structure equations for adapted frames are derived. These equations are
discretized and the geometry of the occuring discrete nets and sphere
congruences is discussed in a conformal setting. This way, the notions of
``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal
systems'' are obtained --- the latter as a generalization of discrete
orthogonal systems in Euclidean space. The relation of a Cauchy problem for
discrete orthogonal nets and a permutability theorem for the Ribaucour
transformation of smooth orthogonal systems is discussed.
|
math
|
865 |
Torus Curves With Vanishing Curvature
|
math.DG
|
Let T be the standard torus of revolution in R^3 with radii b and 1, 0<b<1.
Let \alpha be a (p,q) torus curve on T. We show that there are points of zero
curvature on \alpha for only one value of the variable radius of T,
b=p^2/(p^2+q^2). The curve \alpha has non-vanishing curvature for all other
values of b. Moreover, for this value of b, there are exactly q points of zero
curvature on \alpha.
|
math
|
866 |
On Gromov's theory of rigid transformation groups: A dual approach
|
math.DG
|
Geometric problems are usually formulated by means of (exterior) differential
systems. In this theory, one enriches the system by adding algebraic and
differential constraints, and then looks for regular solutions.
Here we adopt a dual approach, which consists to enrich a plane field, as
this is often practised in control theory, by adding brackets of the vector
fields tangent to it, and then, look for singular solutions of the obtained
distribution. We apply this to the isometry problem of rigid geometric
structures.
|
math
|
867 |
Parallel spinors and holonomy groups on pseudo-Riemannian spinmanifolds
|
math.DG
|
We describe the possible holonomy groups of simply connected irreducible
non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel
spinors.
|
math
|
868 |
Lagrangian two-spheres can be symplectically knotted
|
math.DG
|
This paper shows that there are symplectic four-manifolds M with the
following property: a single isotopy class of smooth embedded two-spheres in M
contains infinitely many Lagrangian submanifolds, no two of which are isotopic
as Lagrangian submanifolds. The examples are constructed using a special class
of symplectic automorphisms ("generalized Dehn twists"). The proof uses Floer
homology.
Revised version: one footnote removed, one reference added
|
math
|
869 |
Symplectic automorphisms of T^*S^2
|
math.DG
|
Let M be the cotangent bundle of S^2, with the standard symplectic structure.
By adapting an argument of Gromov we determine the weak homotopy type of the
group S of those symplectic automorphisms of M which are trivial at infinity.
It turns out that S is weakly homotopy equivalent to \Z. \pi_0(S) is generated
by the class of the standard "generalized Dehn twist". As a consequence, we
show that there are different connected components of S which lie in the same
connected component of the corresponding group of diffeomorphisms.
|
math
|
870 |
On the group of symplectic automorphisms of $\C P^m \times \C P^n$
|
math.DG
|
Let M be the product of \C P^m and \C P^n, with the standard integral
symplectic form. We prove that the inclusion map from the group of symplectic
automorphisms of M to its diffeomorphism group is not surjective on homotopy
groups. More precisely, it is not surjective on \pi_j for all odd j \leq
\max\{2m-1,2n-1\}. This is a weak higher-dimensional analogue of Gromov's
results for \C P^1 \times \C P^1. The proof uses parametrized Gromov-Witten
invariants in a new (?) way. We also give some information about the symplectic
automorphism groups of M with differently weighted product symplectic
structures.
|
math
|
871 |
On the Noncommutative Geometry of the Endomorphism Algebra of a Vector Bundle
|
math.DG
|
In this letter we investigate some aspects of the noncommutative differential
geometry based on derivations of the algebra of endomorphisms of an oriented
complex hermitian vector bundle. We relate it, in a natural way, to the
geometry of the underlying principal bundle and compute the cohomology of its
complex of noncommutative differential forms.
|
math
|
872 |
Twistor spinors on Lorentzian symmetric spaces
|
math.DG
|
We solve the twistor equation on all indecomposable Lorentzian symmetric
spaces explicity.
|
math
|
873 |
Lorentzian twistor spinors and CR-geometry
|
math.DG
|
We prove that there exist global solutions of the twistor equation on the
Fefferman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension
and we study their properties.
|
math
|
874 |
Dolbeault Cohomology of compact Nilmanifolds
|
math.DG
|
Let $M= G/\Gamma$ be a compact nilmanifold endowed with an invariant complex
structure. We prove that, on an open set of any connected component of the
moduli space ${\cal C} ({\frak g})$ of invariant complex structures on $M$, the
Dolbeault cohomology of $M$ is isomorphic to the one of the differential
bigraded algebra associated to the complexification $\cg^\C$ of the Lie algebra
of $G$. To obtain this result, we first prove the above isomorphism for compact
nilmanifolds endowed with a rational invariant complex structure. This is done
using a descending series associated to the complex structure and the Borel
spectral sequences for the corresponding set of holomorphic fibrations. Then we
apply the theory of Kodaira-Spencer for deformations of complex structures.
|
math
|
875 |
On the Cappell-Lee-Miller glueing theorem
|
math.DG
|
We formulate a more conceptual interpretation of the Cappell-Lee-Miller
glueing/splitting theorem using the new language of asymptotic maps and
asymptotic exactness. Additionally, we present an asymptotic description of the
Mayer-Vietoris sequence naturally associated to the Cech cohomology of the
sheaf of local solutions of a Dirac type operator. We discuss applications to
eigenvalue estimates, approximation of obstruction bundles and glueing of
determinant line bundles frequently arising in gauge theoretic problems. The
operators involved in all these results need not be translation invariant.
|
math
|
876 |
Fukaya Floer homology of $Σ\times S^1$ and applications
|
math.DG
|
We determine the Fukaya Floer homology of the three-manifold which is the
product of a Riemann surface of genus $g\geq 1$ times the circle. This sets up
the groundwork for finding the structure of the Donaldson invariants of
four-manifolds not of simple type in the future. We give the following
applications: 1) We show that every four-manifold with $b^+>1$ is of finite
type. 2) Some results relevant to Donaldson invariants of connected sums along
surfaces. 3) We find the invariants of the product of two Riemann surfaces both
of genus greater or equal than one.
|
math
|
877 |
Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature
|
math.DG
|
The following version of a conjecture of Fischer-Colbrie and Schoen is
proved: If M is a complete Riemannian 3-manifold with nonnegative scalar
curvature which contains a two-sided torus S which is of least area in its
isotopy class then M is flat. This follows from a local version derived in the
paper.
|
math
|
878 |
Signatures and Higher Signatures of $S^1$-Quotients
|
math.DG
|
We define and study the signature, A-hat genus and higher signatures of the
quotient space of an $S^1$-action on a closed oriented manifold. We give
applications to questions of positive scalar curvature and to an Equivariant
Novikov Conjecture.
|
math
|
879 |
The Gaussian Measure On Algebraic Varieties
|
math.DG
|
We prove that the ring $\Aff{\R}{M}$ of all polynomials defined on a real
algebraic variety $M\subset\R^n$ is dense in the Hilbert space
$L^2(M,e^{-|x|^2}\de\mu)$, where $\de\mu$ denotes the volume form of $M$ and
$\de\nu=e^{-|x|^2}\de\mu$ the Gaussian measure on $M$.
|
math
|
880 |
Chern classes of modular varieties
|
math.DG
|
Let X be a Hermitian locally symmetric space. We prove that every Chern class
of X has a canonical lift to the cohomology of the Baily- Borel-Satake
compactification X* of X and that the resulting Chern numbers satisfy the
Hirzebruch proportionality formula with respect to the compact dual X^ of X.
The same result holds for any automorphic vector bundle over X in place of the
tangent bundle. As a consequence there is a surjection of the subalgebra of
H*(X*) generated by these lifted classes onto H*(X^). The method of proof is to
construct fiberwise flat connections on these bundles near the singular strata
of X*, where one then finds de Rham representatives of the Chern classes which
are pulled back from the strata.
|
math
|
881 |
Equivariant Cohomology and Wall Crossing Formulas in Seiberg-Witten Theory
|
math.DG
|
We use localization formulas in the theory of equivariant cohomology to
rederive the wall crossing formulas of Li-Liu and Okonek-Teleman for
Seiberg-Witten invariants.
|
math
|
882 |
Hodge theory and cohomology with compact supports
|
math.DG
|
This paper constructs a Hodge theory of noncompact topologically tame
manifolds $M$. The main result is an isomorphism between the de Rham cohomology
with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut
Laplacian $\lap_\mu$ associated to a measure $d\mu$ which has sufficiently
rapid growth at infinity on $M$. This follows from the construction of a space
of forms associated to $\lap_\mu$ which satisfy an ``extension by zero''
property. The ``extension by zero'' property is proved for manifolds with
cylindrical ends possessing gaussian growth measures.
|
math
|
883 |
Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projective differential geometry
|
math.DG
|
It is demonstrated that the stationary Veselov-Novikov (VN) and the
stationary modified Veselov-Novikov (mVN) equations describe one and the same
class of surfaces in projective differential geometry: the so-called
isothermally asymptotic surfaces, examples of which include arbitrary quadrics
and cubics, quartics of Kummer, projective transforms of affine spheres and
rotation surfaces. The stationary mVN equation arises in the Wilczynski
approach and plays the role of the projective "Gauss-Codazzi" equations, while
the stationary VN equation follows from the Lelieuvre representation of
surfaces in 3-space. This implies an explicit Backlund transformation between
the stationary VN and mVN equations which is an analog of the Miura
transformation between their (1+1)-dimensional limits.
|
math
|
884 |
Surfaces with flat normal bundle: an explicit construction
|
math.DG
|
An explicit construction of surfaces with flat normal bundle in the Euclidean
space (unit hypersphere) in terms of solutions of certain linear system is
proposed. In the case of 3-space our formulae can be viewed as the direct Lie
sphere analog of the generalized Weierstrass representation of surfaces in
conformal geometry or the Lelieuvre representation of surfaces in the affine
space.
An explicit parametrization of Ribaucour congruences of spheres by three
solutions of the linear system is obtained. In view of the classical Lie
correspondence between Ribaucour congruences and surfaces with flat normal
bundle in the Lie quadric this gives an explicit representation of surfaces
with flat normal bundle in the 4-dimensional space form of the Lorentzian
signature. Direct projective analog of this construction is the known
parametrization of W-congruences by three solutions of the Moutard equation.
Under the Pl\"ucker embedding W-congruences give rise to surfaces with flat
normal bundle in the Pl\"ucker quadric.
Integrable evolutions of surfaces with flat normal bundle and parallels with
the theory of nonlocal Hamiltonian operators of hydrodynamic type are discussed
in the conclusion.
|
math
|
885 |
Invariant local twistor calculus for quaternionic structures and related geometries
|
math.DG
|
New universal invariant operators are introduced in a class of geometries
which include the quaternionic structures and their generalisations as well as
4-dimensional conformal (spin) geometries. It is shown that, in a broad sense,
all invariants and invariant operators arise from these universal operators and
that they may be used to reduce all invariants problems to corresponding
algebraic problems involving homomorphisms between modules of certain parabolic
subgroups of Lie groups. Explicit application of the operators is illustrated
by the construction of all non-standard operators between exterior forms on a
large class of the geometries which includes the quaternionic structures.
|
math
|
886 |
A traditional dealing with a semi-classical limit and Hopf theorem
|
math.DG
|
This paper deals with a semi-classical limit (Theorem 1) by using traditional
mathematical methods, and shows a Hopf theorem as a corollary. A formal
discussion of it may be found in [7].
|
math
|
887 |
Invariant Forms and Automorphisms of Locally Homogeneous Multisymplectic Manifolds
|
math.DG
|
It is shown that the geometry of locally homogeneous multisymplectic
manifolds (that is, smooth manifolds equipped with a closed nondegenerate form
of degree > 1, which is locally homogeneous of degree k with respect to a local
Euler field) is characterized by their automorphisms. Thus, locally homogeneous
multisymplectic manifolds extend the family of classical geometries possessing
a similar property: symplectic, volume and contact. The proof of the first
result relies on the characterization of invariant differential forms with
respect to the graded Lie algebra of infinitesimal automorphisms, and on the
study of the local properties of Hamiltonian vector fields on locally
multisymplectic manifolds. In particular it is proved that the group of
multisymplectic diffeomorphisms acts (strongly locally) transitively on the
manifold. It is also shown that the graded Lie algebra of infinitesimal
automorphisms of a locally homogeneous multisymplectic manifold characterizes
their multisymplectic diffeomorphisms.
|
math
|
888 |
Eta invariants of Dirac operators on Circle bundles over Riemann surfaces and virtual dimensions of finite energy Seiberg-Witten moduli spaces
|
math.DG
|
We compute eta invariants of various Dirac type operators on circle bundles
over Riemann surfaces via two approaches: an adiabatic approach based on the
results of Bismut-Cheeger-Dai and a direct elementary one. These results,
coupled with some delicate spectral flow computations are then used to
determine the virtual dimensions of Seiberg-Witten finite energy moduli spaces
on any 4-manifold bounding unions of circle bundles. This belated paper should
be regarded as the analytical backbone of dg-ga/9711006. There, we indicated
only what changes are needed to extend the methods of the present paper to
Seifert fibrations and we focused only to topological and number theoretic
aspects related to Froyshov invariants
|
math
|
889 |
Equifocal families in symmetric spaces of compact type
|
math.DG
|
An equifocal submanifold M of a symmetric space N of compact type induces a
foliation with singular leaves on N. In this paper we will show how to
reconstruct the equifocal foliation starting from one of the singular leaves,
the so-called focal manifolds. To be more concrete: The equifocal submanifold
is equal to a partial tube B around the focal manifold and we will show how to
construct B in this paper. Moreover, we will find a geometrical
characterization of focal manifolds.
|
math
|
890 |
L2-torsion of hyperbolic manifolds
|
math.DG
|
The L^2-torsion is an invariant defined for compact L^2-acyclic manifolds of
determinant class, for example odd dimensional hyperbolic manifolds. It was
introduced by John Lott and Varghese Mathai and computed for hyperbolic
manifolds in low dimensions.
In this paper we show that the L^2-torsion of hyperbolic manifolds of
arbitrary odd dimension does not vanish. This was conjectured by J. Lott and W.
Lueck.
Some concrete values are computed and an estimate of their growth with the
dimension is given.
|
math
|
891 |
Semiintegrable almost Grassmann structures
|
math.DG
|
In the present paper we study locally semiflat (we also call them
semiintegrable) almost Grassmann structures. We establish necessary and
sufficient conditions for an almost Grassmann structure to be alpha- or
beta-semiintegrable. These conditions are expressed in terms of the fundamental
tensors of almost Grassmann structures. Since we are not able to prove the
existence of locally semiflat almost Grassmann structures in the general case,
we give many examples of alpha- and beta-semiintegrable structures, mostly
four-dimensional. For all examples we find systems of differential equations of
the families of integral submanifolds V_alpha and V_beta of the distributions
Delta_alpha and Delta_beta of plane elements associated with an almost
Grassmann structure. For some examples we were able to integrate these systems
and find closed form equations of submanifolds V_alpha and V_beta.
|
math
|
892 |
Conformal and Grassmann structures
|
math.DG
|
The main results on the theory of conformal and almost Grassmann structures
are presented. The common properties of these structures and also the
differences between them are outlined. In particular, the structure groups of
these structures and their differential prolongations are found. A complete
system of geometric objects of the almost Grassmann structure totally defining
its geometric structure is determined. The vanishing of these objects
determines a locally Grassmann manifold. It is proved that the integrable
almost Grassmann structures are locally Grassmann. The criteria of
semiintegrability of almost Grassmann structures is proved in invariant form.
|
math
|
893 |
On the theory of almost Grassmann structures
|
math.DG
|
The differential geometry of almost Grassmann structures defined on a
differentiable manifold of dimension n = pq by a fibration of Segre cones SC
(p, q) is studied. The peculiarities in the structure of almost Grassmann
structures for the cases p=q=2; p = 2, q > 2 (or p > 2, q = 2), and p > 2, q >
2 are clarified. The fundamental geometric objects of these structures up to
fourth order are derived. The conditions under which an almost Grassmann
structure is locally flat or locally semiflat are found for all cases indicated
above.
|
math
|
894 |
A conformal differential invariant and the conformal rigidity of hypersurfaces
|
math.DG
|
For a hypersurface V of a conformal space, we introduce a conformal
differential invariant I = h^2/g, where g and h are the first and the second
fundamental forms of V connected by the apolarity condition. This invariant is
called the conformal quadratic element of V. The solution of the problem of
conformal rigidity is presented in the framework of conformal differential
geometry and connected with the conformal quadratic element of V. The main
theorem states:
Let n \geq 4 and V and V' be two nonisotropic hypersurfaces without umbilical
points in a conformal space C^n or a pseudoconformal space C^n_q of signature
(p, q), p = n - q. Suppose that there is a one-to-one correspondence f: V --->
V' between points of these hypersurfaces, and in the corresponding points of V
and V' the following condition holds: I' = f_* I, where f_*: T (V) ---> T (V)
is a mapping induced by the correspondence f. Then the hypersurfaces V and V'
are conformally equivalent.
|
math
|
895 |
Singular points of lightlike hypersurfaces of the de Sitter space
|
math.DG
|
The authors study singular points of lightlike hypersurfaces of the de Sitter
space S^{n+1}_1 and the geometry of hypersurfaces and use them for construction
of an invariant normalization and an invariant affine connection of lightlike
hypersurfaces.
|
math
|
896 |
Upper bounds for the first eigenvalue of the Dirac operator on surfaces
|
math.DG
|
In this paper we will prove new extrinsic upper bounds for the eigenvalues of
the Dirac operator on an isometrically immersed surface $M^2 \hookrightarrow
{\Bbb R}^3$ as well as intrinsic bounds for 2-dimensional compact manifolds of
genus zero and genus one. Moreover, we compare the different estimates of the
eigenvalue of the Dirac operator for special families of metrics.
|
math
|
897 |
On geometry of hypersurfaces of a pseudoconformal space of Lorentzian signature
|
math.DG
|
There are three types of hypersurfaces in a pseudoconformal space C^n_1 of
Lorentzian signature: spacelike, timelike, and lightlike. These three types of
hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed
with a proper conformal structure, and timelike hypersurfaces are endowed with
a conformal structure of Lorentzian type. Geometry of these two types of
hypersurfaces can be studied in a manner that is similar to that for
hypersurfaces of a proper conformal space. Lightlike hypersurfaces are endowed
with a degenerate conformal structure. This is the reason that their
investigation has special features. It is proved that under the Darboux mapping
such hypersurfaces are transferred into tangentially degenerate
(n-1)-dimensional submanifolds of rank n-2 located on the Darboux hyperquadric.
The isotropic congruences of the space C^n_1 that are closely connected with
lightlike hypersurfaces and their Darboux mapping are also considered.
|
math
|
898 |
On a normalization of a Grassmann manifold
|
math.DG
|
On the Grassmann manifold G (m, n) of m-dimensional subspaces of an
n-dimensional projective space P^n, a certain supplementary construction called
the normalization is considered. By means of this normalization, one can
construct the structure of a Riemannian or semi-Riemannian manifold or an
affine connection on G(m, n).
|
math
|
899 |
Teichmuller theory and handle addition for minimal surfaces
|
math.DG
|
We develop Teichmuller theoretical methods to construct new minimal surfaces
in $\BE^3$ by adding handles and planar ends to existing minimal surfaces in
$\BE^3$. We exhibit this method on an interesting class of minimal surfaces
which are likely to be embedded, and have a low degree Gau\ss map for their
genus; the (Weierstrass data) period problem for these surfaces is of arbitrary
dimension.
In particular, we exhibit a two-parameter family of complete minimal surfaces
in the Euclidean three-space $\BE^3$ which generalize the breakthrough minimal
surface of C. Costa; these new surfaces are embedded (at least) outside a
compact set, and are indexed (roughly) by the number of ends they have and
their genus. They have at most eight self-symmetries despite being of
arbitrarily large genus, and are interesting for a number of reasons. Moreover,
our methods also extend to prove that some natural candidate classes of
surfaces cannot be realized as minimal surfaces in $\BE^3$. As a result of both
aspects of this work, we obtain a classification of a family of surfaces as
either realizable or unrealizable as minimal surfaces.
|
math
|
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