**13**

votes

**1**answer

336 views

### Higher Cerf Theory

Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that ...

**1**

vote

**0**answers

109 views

### The relation between Gromov hausdorff convergence and inverse limit of compact metric spaces [closed]

The relation between Gromov hausdorff convergence and inverse limit of compact metric spaces.

**2**

votes

**1**answer

256 views

### Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question:
Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification
$$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$
between the space of symmetric order ...

**8**

votes

**1**answer

192 views

### Dubins car shortest paths: Decidable?

A Dubins car follows a
Dubins path
in $\mathbb{R}^2$, with constant wheel speed and
limited turning radius.
It is known that the shortest Dubins path in the absence
of obstacles follows circular
arcs ...

**1**

vote

**0**answers

143 views

### A simple question in Hitchin's paper “The Geometry of Three-forms in Six Dimensions”

I am reading Hitchin's beautiful paper "The Geometry of Three-forms in Six Dimensions".
Everything goes smooth up to now except for a tiny problem in Section 6.2, which can be formulated as follows. ...

**11**

votes

**3**answers

1k views

### (Very) High dimensional manifolds

Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...

**4**

votes

**1**answer

286 views

### Smooth and $GL(n)$-equivariant implies algebraic?

Context: Let $B_n$ be the space of symmetric bilinear forms on $\mathbb{R}^n$ and $L_n\subset B_n$ be the subset of non-degenerate forms of Lorentzian signature $(-,+,\ldots,+)$. Let $T$ be a finite ...

**5**

votes

**0**answers

178 views

### Noncommutative geometry and line length

I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds ...

**3**

votes

**1**answer

327 views

### The complex structure on $S^{2n}$

Assuming that there is a complex structure on $S^{2n}$ and it
becomes a complex manifold, also assuming there are complex
coordinate $z, w$ on $U, V$ respectively, where $U, V$ are open
cover of ...

**3**

votes

**0**answers

279 views

### How to estimate $|\nabla T|^2\geq c|\delta T|^2$ besides Cauchy-Schwarz inequality?

Let $T$ be a $(0,k)$-tensor on a Riemannian manifold. I was wondering how to improve the estimate $|\nabla T|^2\geq c|\delta T|^2$ for special metrics besides Cauchy-Schwarz inequality?
The question ...

**0**

votes

**2**answers

188 views

### Gradient Ricci soliton

I am reading Cao and Chen's paper "On Bach-flat gradient shrinking Ricci solitons".
A complete Riemannian manifold $(M^n,g_{ij})$ is called a gradient shrinking Ricci soliton if there exists a smooth ...

**2**

votes

**1**answer

182 views

### Is the centralizer $Z_G(A)=\{g\in G| a g= g a\}$ of a finite $A\subset G$ connected for a connected compact Lie group?

Let $G$ be a connected compact Lie group, consider the left/right action on itself.
For any finite $A\subset G$, consider the centralizer
$Z_G(A):=\{g\in G| a g= g a\}$.
Q: is $Z_G(A)$ a connected ...

**3**

votes

**1**answer

207 views

### Dirichlet polyhedra for hyperbolic manifolds

Let $H$ be a simply-connected, complete
space of constant negative curvature, that
is, a hyperbolic space, $\Gamma$ a discrete group of
isometries, and and $M=H/\Gamma$ its quotient space;
we assume ...

**3**

votes

**1**answer

130 views

### Factors of automorphy from Chern connection

This question is inspired by a recent question about holomorphic bundles and factors of automorphy. Suppose $X$ is a compact, complex manifold whose universal cover $\widetilde{X}$ is Stein (the ...

**2**

votes

**0**answers

265 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...

**1**

vote

**0**answers

94 views

### Riemann curvature of $S^1$-principal bundle

Let $(M,g)$ be a Riemannian manifold and $\pi:P \to M$ be $S^1$- principal fiber bundle endowed with a connection $\Gamma$. For every $p\in P$ we have,
$$T_pP \simeq T_pV\oplus\Gamma_p$$
Where $V$ ...

**2**

votes

**0**answers

118 views

### Questions about transformation or integral transformation

I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...

**1**

vote

**1**answer

115 views

### Affine differential geometry. Is Calabi's hypersurface isotropic?

I am in the framework of (equi)affine differential geometry. Let $E$ be a centro-equiaffine space, that is a real vector space of dimension $n$, together with the special linear group $SL_n(R)$. Let ...

**20**

votes

**1**answer

626 views

### Does the Pfaffian have a geometric meaning?

While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph:
"
...In a certain sense, this might be considered a very satisfactory generalization of Guass-Bonnet. ...

**7**

votes

**1**answer

731 views

### Did differential geometry undergo a notation change?

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by ...

**2**

votes

**0**answers

137 views

### Finding a smooth 6-manifold with a closed 2-form which is degenerate only along some embedded 2-spheres

Given a symplectic 6-manifold $(M,\omega)$ and an embedded
symplectic 2-sphere $C\subset M$ whose normal bundle has the first
Chern class -2. How to find on $M$ another closed 2-form $\eta$ which
only ...

**9**

votes

**1**answer

220 views

### Isotropic Riemannian manifolds

Let $M$ be a Riemannian manifold and $G$ a closed connected subgroup of isometries of $M$. Call the pair $(M,G)$ an isotropic pair if $G$ acts transitively on the sphere bundle $SM$. As an example, ...

**3**

votes

**0**answers

135 views

### different proofs of the fact that compact riemann surface has a non-trivial meromorphic function

I would have like to have a list of different proofs of the fact that compact riemann surface has a non-trivial meromorphic function.This is certainly one of the main results of compact riemann ...

**0**

votes

**0**answers

58 views

### Localization on orbit type submanifolds (generalization of Atiyah-Bott-Berline-Vergne)

In equivariant cohomology, the Atiyah-Bott-Berline-Vergne localization theorem says roughly speaking that the integral of an equivariant cohomology class on the $G$-manifold $M$ has only contributions ...

**4**

votes

**0**answers

158 views

### Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...

**5**

votes

**2**answers

382 views

### Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?

There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...

**0**

votes

**0**answers

114 views

### Does exterior product commute functor Hom?

Let $M$ be an module over the commutative ring $R$. I'd like to ask do we have the following isomorphism?
$$Hom_R(\wedge^n_RM,R)\simeq \wedge^n_R Hom_R(M,R)$$
We can obviously see it's true for the ...

**2**

votes

**0**answers

64 views

### Normal-like coordinates for weakly differentiable metrics

Let $(M,g)$ be a Riemannian $W^{2,p}$ metric, with $p>n/2$. Thus $g$ is at least continuous. At any point $P\in M$, do there exist local coordinates $x^i$ such that $g$ can be decomposed as $g_{ij} ...

**1**

vote

**1**answer

213 views

### When does a moduli space admit a spin structure?

This is a very vague question.
Is there any example of spin structures on a moduli space? References are requested.
I have vaguely heard that Witten discussed when a sigma model is spin. Somehow I ...

**1**

vote

**1**answer

110 views

### Large and Small Conformal Groups

It's well-known that on a Riemannian manifold $(M,g)$ with dimension larger than 2, the dimension of its conformal group $\text{conf}(M,g)$ is bounded above by ${n+2\choose 2}$. A Riemannian manifold ...

**4**

votes

**0**answers

180 views

### Differential ideals of Pfaffian forms on jet bundles (Integrability)

(I asked this question on math.stackexchange, but got no reaction in several weeks. So, my conclusion is, that it is harder to answer than I thought, and maybe admissible for the attribute 'research ...

**12**

votes

**3**answers

809 views

### Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for ...

**2**

votes

**2**answers

326 views

### Approximation theorem for Anti-Self-Dual Metrics

Rounge's Theorem states that any meromorphic function on a domain inside $\mathbb{C}$ can be approximated (over compact subsets) by a sequence of rational functions (meromorphic functions on ...

**6**

votes

**1**answer

245 views

### Idea and intuition behind Penrose transform

I would like to know what a Penrose transform is, or more precisely, what is it intended to be - I'm interested in ideas, intuition and some examples of application.
My knowledge of differential ...

**1**

vote

**0**answers

62 views

### Branch point and alexandrov embeddedness

This is a question I have asked on mathstackexchange with a bounty but without any answer; it is probably more adapted to mathoverflow:
Let us assume that $\Sigma_n$ is a sequence of topological ...

**2**

votes

**0**answers

100 views

### the algebraic theory of obstruction of a homology theory

In general homology group of a complex is $Ker d/ Im d$(regardless of grading). However, in many case the square $d\circ d$ is not zero. For example, the study of Floer thoery needs A infinity ...

**0**

votes

**1**answer

66 views

### Is the extrinsic metric and intrinsic metric equivalent if second fundamental form is bounded

Let $M$ be a hypersurface in $\mathbb{R}^{n+1}$ with bounded second fundamental form
$|A|\leq C$. Does intrinsic distance satisfy $d_g(p,q)\leq C'|p-q|$, where $C'$ only depends on $C$. Here ...

**4**

votes

**2**answers

160 views

### The necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold

There is a theorem :
1) 2-dim (pseudo-)Riemannian manifold must be local conformal flat;
2) 3-dim (pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.
3) n-dim (n>3) ...

**6**

votes

**1**answer

151 views

### Laplacian eigenfunction $L^p$ norms

Suppose I have a compact surface (possibly with boundary), and consider the eigenfunctions of the Laplacian, normalized so that their $L^2$ norms are $1.$ Is there some general result or conjecture on ...

**2**

votes

**1**answer

272 views

### Do lower dimensional spheres always lie on higher dimensional spheres?

At a certain stage of my research work, I require the following fact to hold true.
A surface $S$ satisfies certain conditions so that it lies on a 4-sphere in $R^{21}$ (I have used the results proved ...

**1**

vote

**0**answers

83 views

### This weaker version of CR-structure: is it studied somewhere

When I study 5-dimensional $\mathcal{N} = 1$ supersymmetry, I came across such structure as follows.
$(R, \kappa, \Phi, M)$ is an almost contact 5-manifold, such that
\begin{equation}
\kappa ...

**0**

votes

**1**answer

78 views

### Extending connections [closed]

Usually, one views the connection $\nabla$ on a vector bundle $E \to M$ as a map $\Gamma(M, E) \to \Gamma(M,T^*M) \otimes \Gamma(M,E)) \simeq \Gamma(M,T^*M\otimes E)$. One can extend this to the ...

**1**

vote

**1**answer

187 views

### Hamiltonian Isotopy class of Lagrangian Submanifold

Let $(X,\omega)$ be a symplectic manifold, $L\subset X$ be a Lagrangian submanifold, $[L]$ denotes the Hamiltonian isotopy class. How to represent $L'\in[L]$ via $L$ (for example, a graph over $L$)? ...

**0**

votes

**1**answer

179 views

### Dimension of Inverse image

Suppose we have a smooth map between two smooth manifolds $f:M→N$ such that $\dim M>\dim N$, and let $p∈N$ be a critical value. Suppose we know that $X:=f^{-1}(p)$ is a differentiable manifold (or ...

**13**

votes

**0**answers

301 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 ...

**14**

votes

**2**answers

610 views

### Vanishing eigenvalues of Jacobian

Let $f: \mathbb{R^2}\to \mathbb{R^2}$ be a Schwartz function. If the eigenvalues of $Df$ vanish everywhere, must $f$ be constant? Does an analogous result hold when we replace $2$ by $n$?
Any ...

**6**

votes

**0**answers

239 views

### Geometric meaning of the black hole horizon

It is widely accepted that the singularity of the Schwarzschild metric at the event horizon is purely an artifact of the coordinates and no physical singularity exists at the horizon. However, as ...

**5**

votes

**1**answer

394 views

### Differential Geometric Aspects of Rubber Bands

What happens, if a rubber band ( of length $l_0$ that has been stretched to length $l_1:=l_0+\Delta l\;$ and brought into the shape of a closed curve in $\mathbb{R}^3$ ) is released and if the only ...

**2**

votes

**3**answers

253 views

### Can the Einstein Field Equations be written as Difference Equations? [closed]

Does anyone know if the Einstein Field equations have ever been written as Difference Equations, and if so does that simplify anything or produce solutions not available in the usual Differential ...

**6**

votes

**1**answer

176 views

### Tensor product of certain Sobolev spaces on non-compact manifolds

Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in ...