Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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28
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0answers
1k views

Minimal volume of 4-manifolds

This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...
26
votes
0answers
3k views

A paper to the question, if the six dimensional sphere is a complex manifold

Hi, for a few days a paper was published on arxiv.org with the title "The six dimensional sphere is a complex manifold": http://arxiv.org/PS_cache/math/pdf/0505/0505634v3.pdf Because I am not able to ...
25
votes
0answers
1k views

Is S^2 x S^4 a complex manifold?

As observed by Calabi a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is ...
18
votes
0answers
1k views

Ricci flat metric on $n$-sphere?

Can you put a Ricci flat metric on the $n$-sphere, $n>4$?
17
votes
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582 views

Are the eigenvalues of the Laplacian of a generic Kähler metric simple?

It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
15
votes
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309 views

Lipschitz constant of a homotopy

Let $n>0$, $\mathbb S^n$ be $n$-sphere and $1\in \mathbb S^n$ be its north pole. A am looking for an example of compact manifold $M$ with a continuous $n$-parameter family off maps $h_x\colon M\to ...
14
votes
0answers
380 views

Does the Ricci flow approach shed light on Poincare conjecture in higher dimension?

I know that the Poincaré conjecture was first proved in dimension ≥ 5, then dimension 4, and finally 3. I'm just curious, does the Ricci flow approach by Perelman shed any light on the high dimension ...
14
votes
0answers
651 views

Milnor's cartography problem

Let $\Omega$ be a round disc of radius $\alpha<\pi/2$ on the standard sphere. It is easy to construct a $(1,\tfrac{\alpha}{\sin\alpha})$-bi-Lipschitz map from $\Omega$ to the plane. Is it true ...
13
votes
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459 views

Applications of Berger's Curvature Estimate

I'm interested in applications of the following estimate of Berger on the Riemann curvature tensor: Let $(M,g)$ be a Riemannian manifold of dimension $n \geq 4$, let $p \in M$, and assume that the ...
13
votes
0answers
795 views

Nontrivial tangent bundle that is diffeomorphic to the trivial bundle

Is there an example of a smooth $n$-manifold $M$ whose tangent bundle is nontrivial as a bundle but is nonetheless (abstractly) diffeomorphic to the trivial bundle $M \times \mathbb{R}^n$? (This ...
12
votes
0answers
194 views

Artin L-function and Zeta function of twisted Dirac operator

If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation $\sigma$ as a flat connection on the curve, then the definition of the Artin ...
12
votes
0answers
251 views

Has Cheeger's 'de Rham cohomology' of metric measure spaces been studied beyond its definition?

In J. Cheeger's 'Differentiability of Lipschitz Functions on Metric Measure Spaces' (Geometric and Functional Analysis, 1999, Vol. 9 pp 428-517, see here), a 'de Rham cohomology group' ...
11
votes
0answers
306 views

Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...
11
votes
0answers
298 views

What is the holomorphic sectional curvature?

Let $X$ be an $n$-dimensional complex manifold, let $\omega$ be a Kahler metric on $X$ and let $R$ be the $(4,0)$ curvature tensor of $\omega$. We can simplify the tensor $R$ in different ways, two of ...
10
votes
0answers
178 views

A simple proof that parallelizable oriented closed manifolds are oriented boundary?

So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...
10
votes
0answers
389 views

How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?

Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...
10
votes
0answers
288 views

Examples of Clifford Modules

For a Riemannian manifold $M$, a Clifford module is a bundle over $M$ that is, fiberwise, a representation of the Clifford algebra bundle $Cl(TM)$ and has a connection compatible with this action. ...
10
votes
0answers
190 views

“Small” maps from sphere to sphere

Start with a continuous map $f:S^{n+k} \rightarrow S^n$ (round unit spheres). The graph of $f$ lives in $S^{n+k}\times S^n$ and suppose it has a surface area (as a subspace of co-dimension $n$). Now ...
10
votes
0answers
374 views

Exotic smoothness and Parallelizability

Regarding the parallelizability of the Milnor's seven dimensional exotic spheres: Parallelizability of the Milnor's exotic spheres in dimension 7 The following question naturally arises: Suppose ...
9
votes
0answers
115 views

Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates ...
9
votes
0answers
245 views

The cones for Bochner–Lichnerowicz–Weitzenböck formula

The Bochner–Lichnerowicz–Weitzenböck formula can be written the following way $$ \Delta \phi-\nabla^*\nabla \phi= R(\phi),$$ here $\phi$ is a section in a Dirac bundle and $R$ the something which can ...
9
votes
0answers
361 views

Poincaré recurrence and symplectic packings

Question. Is there any example of a path connected symplectic manifold $(M,\omega)$ that has infinite volume, but which cannot be packed by an infinite number of symplectic balls of a fixed radius ...
9
votes
0answers
286 views

3 manifolds with diffeomorphic unit tangent bundles

What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?
9
votes
0answers
418 views

What is Quillen's contribution to index theorem?

In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...
9
votes
0answers
146 views

When is a submersion locally volume-expanding?

I would like to characterize the smooth maps $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^k$, $n\geq k$, with the following property: For every $x\in \mathbb{R}^n$ there exists a positive number ...
9
votes
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396 views

Killing spinors and symmetric tensor fields.

Hi all, I have a question of the following form: Let $(M,g)$ be a Riemannian spin manifold which admits a Killing spinor $\sigma$ and let $h:T M \to T M$ be a symmetric, trace-free and ...
9
votes
0answers
313 views

Covering a subset by submanifolds

Let $K$ be a compact subset of a Euclidean space of very large dimension $N$. Assume that any point $x\in K$ has a neighborhood $U\subset K$ which is contained in a smooth $l$-dimensional submanifold ...
8
votes
0answers
233 views

Carleson's Theorem on Manifolds

Let $M$ be an oriented, compact, differentiable manifold with some Riemmanian metric $g$, so that $(M,g)$ has a nice volume form and one can define $L^2(M,g)$ as the completion of $C^\infty(M)$ under ...
8
votes
0answers
429 views

Seemingly elementary geometric problem in R^3 which requires the axiom of choice

While playing with what I called "quantum matching", the following problem arose: construct a map $F$ from the unit sphere $S_2$ in $R^3$ to itself such that $F(X)$ is orthogonal to $X$ plus has one ...
8
votes
0answers
128 views

What sort of geometry does the Whitehead manifold have as a hypersurface in $\mathbb{R}^4$?

If I understand correctly, the standard $\mathbb{R}^4$ is diffeomorphic to $\mathbb{R}\times W$ where $W$ is the Whitehead manifold (i.e., is an open three-manifold that is contractible and not ...
8
votes
0answers
217 views

Are the Chern numbers of a hyperbolic-type compact complex manifold bounded in terms of the Euler number?

Let $X$ be an $n$-dimensional compact Kahler manifold with negative first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i \geq 0$ with $\sum ik_i = n$, bounded in terms ...
8
votes
0answers
364 views

non-vanishing magnetic helicty density

Suppose you are given a nowhere-vanishing exact 2-form $B=dA$ on an open, connected domain $D\subset\mathbb{R}^3$. I'd like to think of $B$ as a magnetic field. Consider the product $H(A)=A\wedge ...
8
votes
0answers
217 views

Embeddings of hyperbolic $n$-manifolds in $R^{n+2}$

Is there any example of a compact manifold $M$ of dimension $n>10000$ such that $M$ admits an embedding into $\mathbb R^{n+2}$, $M$ is hyperbolic; i.e., it admits a Riemannian metric with ...
8
votes
0answers
199 views

Exhaustion of an open manifold of bounded curvature and finite volume

In the Cheeger-Gromov paper "On the Characteristic Numbers of Complete Manifolds of Bounded Curvature and Finite Volume", http://www.maths.ed.ac.uk/~aar/papers/cheegergr1.pdf, the authors make the ...
8
votes
0answers
391 views

How does duality of symmetric spaces explain the hyperbolic cosine theorem?

There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $G/K$ is a symmetric space of noncompact type, $g=k+p$ the ...
8
votes
0answers
558 views

Moduli space of semistable bundles

It is well-known that the space of S-equivalence classes of rank 2 semistable holomorphic vector bundles with trivial determinant on a genus 2 surface M is $CP^3$ (more concretely ...
8
votes
0answers
404 views

What is the origin of the formula for the Lie derivative along a Killing vector?

Background Let $(M,g)$ be an $n$-dimensioal riemannian manifold. A vector field $X$ on $M$ is said to be a Killing vector if the flow it generates is an isometry; that is, it preserves the metric ...
8
votes
0answers
494 views

Hausdorff measure question

Say we have some compact metrisable topological space $X$ with a measure $\mu$ defined on the Borel sets of $X$. Then is there some way to determine whether $\mu$ is the Hausdorff measure associated ...
7
votes
0answers
227 views

diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...
7
votes
0answers
79 views

Euler number of the complex of basic forms

Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action. I am trying to understand the Hopf bundle ...
7
votes
0answers
196 views

quantitative version of the rigidity of the 2-sphere

I am looking for a quantitaive version of the following theorem: A compact surface with $K\equiv 1$ is isometric to the round sphere. Of course I get the Berger, Brendle-Schoen Theorem which insures ...
7
votes
0answers
197 views

Surfaces with many (but not solely) closed geodesics?

Let $S$ be a closed surface embedded in $\mathbb{R}^3$, let's say of genus zero. I seek examples of $S$ with the following property: If one selects a random any point $p$ on $S$, and a random ...
7
votes
0answers
268 views

Curvature as infinitesimal holonomy

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy ...
7
votes
0answers
175 views

Systoles of hyperbolic (Riemann) surfaces of large genus

Let $m$ be a Riemannian metric on $S_g$ the surface of genus $g$, and $sys(m)$ be the length of the shortest non contractible cycle with respect to $m$. The systolic inequality claims that for any ...
7
votes
0answers
151 views

Immersions of Topoi

An immersion of smooth manifolds is a smooth map whose Jacobian has full rank at each point in the source manifold. Is there a notion of ``immersion'' for geometric morphisms of topoi which ...
7
votes
0answers
340 views

Are there exotic $S^2\times S^2$?

On 2010 AKHMEDOV and PARK claimed there are infinitely many exotic smooth structures on $S^2\times S^2$, see http://arxiv.org/abs/1005.3346 Then Rasmussen posted a paper : Perfect Morse functions and ...
7
votes
0answers
207 views

What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...
7
votes
0answers
258 views

Why does the index of the Dirac operator on a manifold with boundary live inside the Pfaffian line of the boundary Dirac operator?

I am trying to understand a statement from page 72 of What is an elliptic object? by Stolz and Teichner. They have a spin Riemannian 2-manifold $\Sigma$, with boundary 1-manifold $Y$. The Dirac ...
7
votes
0answers
546 views

Good introduction to Morse-Novikov theory?

Morse theory investigates the topology of compact manifolds using critical points of real-valued functions $f\colon\, M\to \mathbb{R}$. Motivated by problems in dynamical systems, Novikov (Multivalued ...
7
votes
0answers
307 views

Computing the Chern-Simons invariant of $SO(3)$

I recently tried asking this question on math.stackexchange.com but I have not yet received any feedback so I thought I should try asking here. I apologize if this question is too basic. The question ...