**14**

votes

**1**answer

331 views

### The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g−6$ geodesic length functions

I asked this question almost a month ago on Math SE. After waiting three weeks for an answer or a comment, I opened a bounty on the question in hope that it might get an answer this way. The bounty ...

**13**

votes

**1**answer

420 views

### Can a Riemannian metric be analytic in non-analytically different coordinates?

Suppose I have two coordinates on the same (subset of a) Riemannian manifold.
If the metric tensor is analytic in both coordinates, is the change of variables between them necessarily analytic?
In ...

**1**

vote

**1**answer

144 views

### stable bundle on Calabi-Yau 3-fold

Let $E$ be a $G$-bundle over a compact Calabi-Yau 3-fold, $G$ is semi-simple,compact Lie group. If $E$ is a stable bundle, is the first Chern class of $E$ always zero?

**4**

votes

**2**answers

149 views

### Recover Embedding from Metric

Suppose that $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$; the embedding is not explicitly known.
And suppose that I know ...

**3**

votes

**1**answer

117 views

### Kunneth formula for de Rham cohomology twisted by flat vector bundle

It is well known that for two manifolds $M$ and $N$ (let's say they are compact), the Kunneth formula says that $H(M\times N)=H(M)\otimes H(N)$, where $H$ denotes de Rham cohomology with complex ...

**3**

votes

**2**answers

168 views

### Associated vector bundles and Characteristic Classes

Assume that $P\to M$ is a principal $G$-bundle where $G$ is some (compact) Matrix group. Let $\rho\colon G \to \operatorname{Gl}(\mathbb{R}^n)$ be the tautological representation and $\rho^\prime\...

**2**

votes

**1**answer

110 views

### Carathéodory's theorem for $SO(3)$?

Let $Q \in \operatorname{Conv} SO(3)$.
Is there a way to retrieve an explicit representation of $Q$ as convex combination $Q=\sum_{k=1}^{r}{\lambda_{i}R_{i}},
R_{i} \in SO(3)$?
An approximation ...

**2**

votes

**0**answers

140 views

### Negative-curvature behaviour of higher-rank symmetric spaces

Let $X$ be a symmetric space of noncompact type with $rk\ X\geq 2$ and let $G$ be the identity component of the isometry group. Pick points $p\in X$ and $\xi\in\partial_{\infty}X$, with $\xi$ regular, ...

**0**

votes

**0**answers

91 views

### Definition of formal adjoint of covariant derivative [migrated]

I read in Einstein Manifolds, L. Besse that the covariant derivative $D: \mathcal{J}^{(r,s)}(M)\to \Omega^1(M)\otimes\mathcal{J}^{(r,s)}(M)$ admit an formal adjoint $D^*:\Omega^1(M)\otimes\mathcal{J}^{...

**0**

votes

**0**answers

183 views

### Can some exotic sphere be diffeomorphically embedded into some $R^n$?

Can some (or perhaps every) exotic sphere be diffeomorphically embedded into some $R^n$? How does such an embedding (if it exists) look like? I.e., what are the equations for a particular embedding?
...

**1**

vote

**0**answers

105 views

### Ellipticity of Bott-Chern Laplacian

I want to prove that Bott-Chern Laplacian
$$\tilde{\Delta}_{BC}^{p,q}=(\partial\bar\partial)(\partial\bar\partial)^*+(\partial\bar\partial)^*(\partial\bar\partial)+(\bar\partial^*\partial)(\bar\...

**0**

votes

**1**answer

76 views

### Exterior derivative on principal bundle [closed]

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...

**5**

votes

**2**answers

551 views

### Can a Morse function define a unique structure on a closed manifold?

I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have ...

**1**

vote

**0**answers

111 views

### Name for the Quotient $SU(m+1)/(SU(k) \times SU(m-k))$

The sphere $S^{2m-1} \simeq SU(m+1)/SU(m)$ has a canonical $U(1)$-action, and quotienting by this action give complex projective space $CP^m$. We can generalise the family of sphere to the family of ...

**2**

votes

**1**answer

182 views

### Scheme of Higgs reductions

I'm reading the Bruzzo and Graña Otero's paper Semistable and Numerically Effective Principal (Higgs) Bundles; here: $X$ is a smooth, complex, projective variety; $G$ is a connected, complex, ...

**0**

votes

**1**answer

131 views

### Gaussian bounds on Dirichlet heat kernel

Let $(M, g)$ be a compact Riemannian manifold and let $p(t, x , y)$ be the heat kernel of $M$. Then there exist constants $c, C > 0
$ such that $$\frac{c}{t^{n/2}}
e^{-\frac{1}{4t}d(x, y)^2} \leq ...

**7**

votes

**0**answers

471 views

### Counting limit cycles via curvature in Riemannian geometry

In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem
First we give a short introduction:
A quadratic system is a polynomial vector field on ...

**1**

vote

**1**answer

203 views

### An answer to this system of PDE's

Planning of the question:
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle
The isotropic almost complex structures $J_{\delta , \sigma}$ were introduced by Aguilar on the ...

**4**

votes

**1**answer

218 views

### Hodge decomposition for Bott-Chern cohomology

$\DeclareMathOperator{im}{im}$
I want to prove that Bott-Chern cohomology group $H^{p,q}_{BC}=\frac{\ker\partial\cap \ker\bar{\partial}}{\im\partial\bar{\partial}}$ has finite dimension via Hodge ...

**14**

votes

**3**answers

545 views

### Is SO(2n+1)/U(n) a symmetric space?

I am a physics student with only a rudimentary knowledge of differential geometry, so please feel free to point out if I miss something elementary / trivial.
According to https://arxiv.org/abs/1408....

**5**

votes

**1**answer

188 views

### Question on Weil-Petersson metric on Teichmuller space

I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space".
The tangent space at ...

**2**

votes

**0**answers

47 views

### Second Countability hypothesis for a Banach manifold

Is the second countability hypothesis necessary to rigoruosly define a Banach Manifold (say in infinite dimension)?
In the finite-dimensional theory of manifolds, that request is included in the ...

**1**

vote

**1**answer

252 views

### A Geometric proof of the Gauss Lucas theorem

Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask:
Is there a geometric proof for the Gauss-Lucas theorem ?Since we are working on a half plane, can one imagine a possible ...

**3**

votes

**0**answers

65 views

### Various definitions of the odd Chern character form

I am asking this question from my possibly defected memory, so the things below may not be accurate.
I want to know how many different definitions of the odd Chern character form using differential ...

**0**

votes

**0**answers

61 views

### charts for some riemannian embedding

Let $(M,g)$ be some riemannian manifold with some Lie group $G$ acting properly, freely and by isometries. So $M/G$ is a manifold and using the projection $\pi \colon M \to M/G$, we get a smooth ...

**5**

votes

**0**answers

181 views

### Origin of the name ''momentum map''

Why is the momentum map in the differential geometry of symmetries called the ''momentum'' (or ''moment'') map?

**1**

vote

**1**answer

111 views

### Convergence of Discrete Geodesic

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$.
Suppose $f^{-1}:U_p \mapsto U$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the sequence:
\...

**5**

votes

**1**answer

243 views

### Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on ...

**5**

votes

**1**answer

111 views

### Geometric interpretation of splitting of sequence associated to a homogeneous space

Let $G$ be a Lie group acting transitively on a smooth manifold $M$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and let $\xi : \mathfrak{g} \to \Gamma(TM)$ be the Lie algebra homomorphism sending $...

**2**

votes

**1**answer

78 views

### harmonic differential form integer class

Let $(M,g)$ be a compact Riemannian three-fold such that $H_2(M,\mathbb{Z}) = \mathbb{Z}$ and $S$ any surface representing 1. By Hodge theory there exist a harmonic differential one-form $\eta$ dual ...

**1**

vote

**0**answers

110 views

### Gravitational instantons metric (change variables)

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric:
$$
\gamma dz d\bar{z}+\gamma^{...

**5**

votes

**1**answer

212 views

### On Johansson's Theorem on homotopy equivalences of 3-manifolds

Johansson's theorem states the following:
Given $f:M_1\rightarrow M_2$ (not a pair map) an homotopy equivalence between 3-manifolds with incompressible boundary.
Let $V_i$ be the components of the ...

**5**

votes

**0**answers

97 views

### Holonomy group of a dense, open submanifold

I have a Riemannian manifold $(X,g)$, where $X$ is not necessarily compact or complete, and a dense open submanifold $Y \subseteq X$. In my case $X$ is a smooth quasi-projective variety over the ...

**1**

vote

**0**answers

86 views

### Restriction of real analytic functions to embedded submanifolds

lets assume we have a real vectorspace $V$ and functions $f_1, \dots, f_k \colon V \to \mathbb{R}$ which are real-analytic (for instance, let them be polynomial).
Furthermore we have an embedded real-...

**6**

votes

**1**answer

160 views

### Geometric quantization of Teichmuller space

The quantizations of Teichmuller space I have seen are via special coordinates (e.g. the paper of Chekhov and Fock) or conformal blocks. Does one get an equivalent quantization by geometric ...

**1**

vote

**1**answer

89 views

### Curvature of plane curves on a surface

Let $S$ be a surface and $\gamma$ a curve on $S\subseteq \mathbb{R}^3$ obtained cutting $S$ with a plane. I wuold an upper bound for the curvature of $\gamma$. Are there papers for this topic?

**1**

vote

**1**answer

208 views

### Ricci curvature and killing form

Motivated by this question we ask:
Is there any relation between the Ricci curvature of a Lie group and the killing form of its Lie algebra?Under what conditions, they are proportional to each ...

**4**

votes

**1**answer

76 views

### sequence of graphs converge in the sense of varifold to multiplicity 2 plane

Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) ...

**19**

votes

**2**answers

2k views

### Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked
Is it true, as rumours have it, that
you started to work on the embedding ...

**15**

votes

**5**answers

665 views

### Smoothness of the closest point on a submanifold

Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold.
Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...

**16**

votes

**0**answers

505 views

### Is the determinant of cohomology a TQFT?

If $M$ is an oriented $d$-manifold, let $D(M)$ denote the top exterior power of $H^*(M,\mathbf{C})$. Then $D(M_1 \amalg M_2) = D(M_1) \otimes D(M_2)$. Is there a good recipe for a map $D(M) \to D(N)$...

**4**

votes

**1**answer

387 views

### “Polygons and gravitons” and Kodaira's theorem

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 470. At this point, he does some computations and obtains the conformal structure of the real ...

**5**

votes

**0**answers

156 views

### Contact manifolds and pseudodifferential operators

By way of background, I am currently trying to understand the theory of pseudodifferential operators in the context of contact geometry. I have some knowledge of pseudodifferential operators on ...

**6**

votes

**1**answer

151 views

### The radially symmetric isoperimetric problem

A solution to the $l$-isoperimetric problem on a Riemannian surface $(M,g)$ is a smooth closed curve $\gamma \subset M$ of length $l$ which minimizes the isoperimetric constant:
$$h(\gamma) = \frac{l}{...

**2**

votes

**1**answer

53 views

### coisotropic submanifolds on poisson manifolds

Let $(M, \{.,.\})$ be a Poissonmanifold and $B$ the corresponding Poissontensor. Now in this context, a embedded submanifold $C \subset M$ is called coisotropic, if $B^\#(TC^\circ) \subset TC$.
For ...

**5**

votes

**1**answer

194 views

### minimal surfaces in $S^n$

Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:
Can we remove the embeddness assumption?
Can we ...

**5**

votes

**0**answers

85 views

### Relationship between Gaussian and bisectional curvature

Let $f:\mathbb CP^1\to X$ be an smooth embedding which its image is the curve $C$ where $X$ is a Kähler manifold. Do we have that $$\sup_{\mathbb CP^1}K(f^*\omega)<\sup_C K(\omega),$$where $K(f^*\...

**7**

votes

**2**answers

466 views

### Spin^c structures on manifolds with almost complex structure

Let $M$ be a smooth even-dimensional manifold.
Is it true that for each almost-complex structure $J$ on $M$ there exists a canonical spin$^c$ structure $S_J$ associated to $J$ ?
(I've read this ...

**15**

votes

**2**answers

543 views

### Are there some intrinsic invariants of surfaces other than Gaussian curvature?

The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$.
Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...

**8**

votes

**0**answers

217 views

### Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?

The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...