**29**

votes

**0**answers

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 ...

**28**

votes

**0**answers

557 views

### Are there periodicity phenomena in manifold topology with odd period?

The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...

**26**

votes

**0**answers

2k 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 ...

**19**

votes

**0**answers

1k views

**17**

votes

**0**answers

611 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

**0**answers

324 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 ...

**15**

votes

**0**answers

671 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 ...

**14**

votes

**0**answers

397 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 ...

**13**

votes

**0**answers

323 views

### Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms ...

**13**

votes

**0**answers

461 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 ...

**13**

votes

**0**answers

502 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

**0**answers

879 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

**0**answers

259 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

**0**answers

548 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 ...

**12**

votes

**0**answers

497 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 ...

**12**

votes

**0**answers

287 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

**0**answers

185 views

### A variation on the local Günther inequality

This question is about a variation on the Günther (also known as Günther-Bishop) inequality for manifolds of sectional curvature bounded from above. With Greg Kuperberg, we would deduce from it a ...

**10**

votes

**0**answers

165 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
...

**10**

votes

**0**answers

266 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 ...

**10**

votes

**0**answers

311 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 $S^2 \times M_2$?

**10**

votes

**0**answers

308 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

**0**answers

161 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 ...

**10**

votes

**0**answers

201 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

**0**answers

426 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 ...

**10**

votes

**0**answers

389 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

**0**answers

206 views

### Reference for a proof of the fiberwise Stokes theorem

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary,
the difference between the fiberwise integral of the differential and the ...

**9**

votes

**0**answers

378 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

**0**answers

461 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

**0**answers

178 views

### Is there a Hodge isomorphism theorem for part-tangential, part-normal, harmonic differential forms?

Let $M$ be an oriented compact Riemannian $n$-manifold with boundary $\partial M$. A differential $p$-form $\omega$ on $M$ is normal if $i^* \omega = 0$ holds, tangential if $i^* \star \omega = 0$ ...

**9**

votes

**0**answers

378 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 ...

**9**

votes

**0**answers

426 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

**0**answers

318 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

**0**answers

213 views

### What is a good introduction to cluster algebras from surfaces?

What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?
In my view, that means it should start off with unpunctured surfaces (and in ...

**8**

votes

**0**answers

160 views

### Commutative spectral triples not coming from manifolds

There is a very deep and remarkable theorem by Connes (the so called reconstruction theorem) which states that from a commutative spectral triple obeying certain axioms one can reconstruct a smooth ...

**8**

votes

**0**answers

261 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

**0**answers

455 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

**0**answers

132 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

**0**answers

229 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

**0**answers

382 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

**0**answers

226 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

**0**answers

214 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

**0**answers

647 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 ...

**8**

votes

**0**answers

576 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

**0**answers

448 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

**0**answers

511 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

**0**answers

73 views

### Kinematics of rolling knots

It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes.
(An ...

**7**

votes

**0**answers

247 views

### Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?

Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...

**7**

votes

**0**answers

198 views

### Flat manifolds and irreducible representations

Let $M$ be a compact Riemannian manifold with vanishing curvature of Levi-Civita connection. Such manifolds were classified by Bieberbach; sometimes they are called Bieberbach manifolds. According to ...

**7**

votes

**0**answers

228 views

### Reconciling the affine grassmannian and the based loop group

I'm trying to reconcile the differences between the (algebraic) based loop group and the affine grassmannian. I once believed that I understood the relationship, but I just read a paper which has ...

**7**

votes

**0**answers

123 views

### Topological restrictions from mean curvature bounds

Alexandrov's Theorem says that a compact constant mean curvature hypersurface embedded in $\mathbb{R}^{n+1}$ must be a round sphere.
What happens when the mean curvature is small, or bounded? (For ...