**3**

votes

**0**answers

74 views

### Geometry of ends of a finite volume negatively curved manifold

Is there a survey of the geometry of manifolds with finite volume Riemannian metrics of negative sectional curvature? More specifically, I am interested in the geometry of cusp ends of such manifolds, ...

**2**

votes

**1**answer

154 views

### What is the meaning of Yang-Mills action evaluated on Levi-Civita connection?

On a Riemannian manifold $M$ with riemann curvature tensor $R_{\mu\nu\rho\sigma}$ written as (endomorphism valued) curvature two-tensor of the Levi-Civita connection $R=R_{\mu\nu}dx^\mu\wedge ...

**5**

votes

**1**answer

285 views

### A question on generalized Einstein metrics on four-dimensional manifolds

I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds,
\begin{equation*}
...

**0**

votes

**0**answers

46 views

### Guassian upper bound of the heat kernel implies ultracontrativity?

For spaces satisfies uniformlly local doubling and Poincare inequality (for example, Riemannian manifold with Ricci curvature bounded below RCD(K,N)).
By Sturm's paper, we have bounds on the heat ...

**1**

vote

**1**answer

196 views

### Possible homotopy-theoretical approach to Gauss-Bonnet

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...

**1**

vote

**1**answer

231 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 connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least ...

**-4**

votes

**0**answers

20 views

### Relationship for cosine of angle [migrated]

If $x$ is the cosine of the angle between the vectors $a$ and $b$, $y$ is the cosine of the angle between the vectors $a$ and $p$, and $z$ is the cosine of the angle between the vectors $b$ and $p$, ...

**1**

vote

**0**answers

84 views

### Kaehlerian metric on a tangent bundle

I am trying to construct a Kaehler structure on $R^{4}$ following paper of V. Oproiu, "Some new geometric structures on the tangent bundle".
This is how the author constructs a metric on a tangent ...

**16**

votes

**5**answers

797 views

### Isometric embedding of SO(3) into an euclidean space

Consider $SO(3)$ with its bi-invariant metric and $R^n$ the euclidean space of dimension $n$. What is the minimal value of $n$ such that there exists an isometric embedding $f: SO(3) \to R^n$?

**1**

vote

**0**answers

78 views

### Buseman function for Riemanniam manifolds with two ends and $Ric\ge -(n-1)$ [closed]

It's well known that if M is a Riemannian manifold with $Ric \ge 0$ and contains a line $\gamma $.
Set $${\gamma _ + } = \gamma | {_{[0, + \infty )}} ,{\gamma _ - } = \gamma | {_{[ - \infty ,0)}} ...

**1**

vote

**1**answer

194 views

### How does the inverse mean curvature flow start with minimal surface?

I am unclear how the inverse mean curvature flow starts with a minimal surface. If there is some point $p$ with mean curvature $H(P)=0$, how should we treat this problem and let the flow start?

**4**

votes

**2**answers

185 views

### Relative Serre spectral sequences?

I consider the following two situations:
Let $B$ be a simply connected space, and $F\to E\to B$, $F'\to E'\to B$ two fibrations with a map $f:E'\to E$ sending fibers to fibers and inducing the ...

**1**

vote

**0**answers

147 views

### a good introduction to Laplace Beltrami operator over differential manifolds? [migrated]

I'd like to have a good reference to understand how the Laplacian operator get generalized over differential manifolds.
More concretely, I want to understand and prove the equation :
$$\Delta ...

**2**

votes

**1**answer

142 views

### Surface curves equidistant from a simple closed geodesic

Let $S \subset \mathbb{R}^3$ be a surface embedded in $\mathbb{R}^3$,
let's say (to keep it simple) of genus zero.
Let $\gamma$ be a simple, closed, oriented geodesic on $S$.
Because $\gamma$ is ...

**4**

votes

**2**answers

333 views

### Ricci curvature under rough convergence

From the work of Lott--Villani and Sturm, I know that the following fact holds:
(*) Suppose that $(M_k,g_k,dvol_{g_k})$ is a sequence of compact Riemannian manifolds of non-negative Ricci ...

**9**

votes

**0**answers

193 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 ...

**5**

votes

**2**answers

287 views

### Volume-preserving projective transformations are isometries

What is a simple, elementary proof of the following result?
A continuously differentiable map from the unit sphere $S^n \subset \mathbb{R}^{n+1}$ $(n > 1)$ to itself that preserves volumes and ...

**4**

votes

**2**answers

195 views

### Rank of a jet bundle of a vector bundle. Interpretation of the first jet bundle

I am trying to understand the jet bundles but currently I am stuck on the following questions:
Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...

**15**

votes

**1**answer

388 views

### Does the holonomy map define a homomorphism $\pi_k(X)\to\pi_{k-1}(Hol(\nabla))$?

Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...

**3**

votes

**1**answer

119 views

### Fréchet-Kolmogorov compactness Theorem for Lp spaces on manifolds

Suppose I have a family of functions $\mathcal{F} \subseteq L^2(\mathcal{M}, P)$ where $\mathcal{M}$ is a compact manifold, and $P$ is a probability distribution on $\mathcal{M}$. Is there an ...

**1**

vote

**1**answer

172 views

### Existence of positive solutions of a linear PDE on closed manifolds

I was wondering is there a sufficient condition (or sufficient and necessary condition) for the existence of positive solutions of the following linear PDE on a closed manifold $(M, g)$,
...

**1**

vote

**2**answers

222 views

### The Schottky group and the fundamental group of a compact Riemann surface

I am quoting the following description from a paper,
"...every compact Riemann surface can be obtained as the quotient $\mathbb{C}/\Gamma$ where $\Gamma$ is a Schottky group. The Schottky group of a ...

**2**

votes

**2**answers

174 views

### In what condition is a conformal flat manifold flat?

$g^{\mu\nu}(x)=\Omega^{2}(x)g'^{\mu\nu}(x)$ is a conformal transformation.
If $g'^{\mu\nu}$ is flat, what kind of $\Omega(x)$ is choosed can make $g^{\mu\nu}$ flat.
We can think about any dimension ...

**4**

votes

**2**answers

252 views

### Conditions for a curve to belong to a hyper-surface in $\mathbb{R}^n$

I am recently concerned with the following problem: Given a parametrized curve in $\mathbb{R}^6$, what is the condition that it belongs to a hyper-sphere of dimension 5?
The more general question ...

**1**

vote

**0**answers

38 views

### Construction of a Hermitian metric on a locally free subsheaf which is not a subbundle

Assume $X$ is a smooth projective curve over $\mathbb{C}$ and let $\mathcal{E}$ be a locally free sheaf of rank 2 on $X$. We pick a closed point $x\in X$ and a surjection $f: \mathcal{E}\rightarrow ...

**1**

vote

**1**answer

136 views

### A question about the first eigenvalue for two Kahler metrics

While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies ...

**7**

votes

**1**answer

275 views

### Morse Theory and proper actions

Is there some reference containing an ellaboration of Morse Theory for proper actions of Lie groups?

**9**

votes

**1**answer

350 views

### Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles?
Can you give me a text or an article?
For example:
Consider a vector bundle $E$ with fiber $V$ and base manifold $M$.
Consider ...

**5**

votes

**0**answers

67 views

### Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?
Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...

**1**

vote

**0**answers

74 views

### Riemann normal coordiantes and change of metric

Le $(M,g)$ be Riemannian manifold. Fix point $p\in M$. We can define the map
$$\exp: U \subset T_p M \rightarrow M$$
$$\exp(X) = \gamma_{p,X}(1)$$
where $t\mapsto \gamma_{p,X}(t)$ is geodesic such ...

**1**

vote

**1**answer

145 views

### Zeroes of a complex polynomial on a sphere as a manifold

Let $ f \in \mathbb{C}[z_1, \ldots, z_n]$ be a polynomial such that $f'(z) \neq 0$ if $z \neq 0$ ($f'$ means $\left( \frac{\partial f}{\partial z_1}, \ldots, \frac{\partial f}{\partial z_n}\right)$ ). ...

**4**

votes

**1**answer

172 views

### Exercises around Diffeological Spaces or a Diffeologic Atlas Theory

Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...

**4**

votes

**2**answers

140 views

### Lagrangians with the same extremal curves

It is well know that (in the sense having the same image not parametrization) the extremal curves of the energy functional on a Riemanian manifold $(M,g)$:
$E[\gamma(t)] = \int ...

**2**

votes

**1**answer

215 views

### undergraduate handle decomposition. Reference

As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.

**7**

votes

**2**answers

260 views

### Necessary condition on Calabi-Yau manfiold to be a hypersurface in a Fano manifold

Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it
being a member of a base-point-free linear system in a nef-Fano fourfold?
What, in anything, is known ...

**1**

vote

**1**answer

185 views

### Solvable question of dee dee bar lemma

Recently I read about the dee dee bar lemma ($\partial\bar \partial$-lemma) in Gang Tian's Canonical metrics in Kähler Geometry. In the middle of Page 16, the author writes that: "The following ...

**1**

vote

**3**answers

223 views

### Two questions on isometric embedding

According to the answer of the following question, I try a new version:
An special isometric embedding
Let $M$ be a Riemannian manifold and $\gamma$ be a small part of a geodesic.
Is there an ...

**0**

votes

**1**answer

71 views

### An special isometric embedding

Let $M$ be a Riemannian manifold and $\gamma$ be a non closed geodesic.
Is there an isometric embedding of $M$ into some $\mathbb{R}^{n}$ which send $\gamma$ into an straight line?
The second ...

**3**

votes

**1**answer

182 views

### What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence?

If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, ...

**1**

vote

**1**answer

231 views

### A question about complex polarization

Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar P ...

**4**

votes

**1**answer

330 views

### Triple bubble conjecture: Natural candidate?

Is there a standard natural candidate surface for
the shape that encloses three given volumes
in $\mathbb{R}^3$ and has minimal surface area?
I know the planar triple bubble conjecture was ...

**0**

votes

**0**answers

35 views

### Existence of local frames in Hilbert manifolds

Thank you for looking at my queation.
I'm studying about Hilbert manifolds. I would like to ask you if there exists some concept like a local frame for every Hilbert manifold.
I'm refering to the ...

**13**

votes

**2**answers

426 views

### How useful/pervasive are differential forms in surface theory?

Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...

**10**

votes

**0**answers

150 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 ...

**1**

vote

**1**answer

86 views

### Cut locus, conjugate points and smoothness of distance function

I had a couple of related questions on the cut locus, conjugate points and smoothness of distance function. Let $(M,g)$ be a smooth complete Riemannian manifold and $r(x) = d(p,x)$ the distance ...

**2**

votes

**1**answer

161 views

### almost holomorphic line bundles

Let $(L,\omega_L) \to (M,\omega_M)$ be a symplectic line bundle (symplectic line=real dimension 2) over a symplectic manifold $M$. Each of these objects can be equipped with an almost complex ...

**3**

votes

**1**answer

175 views

### Newlander-Nirenberg in dimension 2

What is the easiest (and what is the most elementary) way of proving
Newlander-Nirenberg theorem for Riemannian surfaces? I was able to reduce
it to existence of non-trivial harmonic functions ...

**7**

votes

**1**answer

203 views

### Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)

Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in ...

**0**

votes

**0**answers

27 views

### When is Dirichlet solution from disk to Riemannian manifold Holder continuous near the boundary?

$D$ is the two-dimensional unit disk. $X$ is a compact Riemannian manifold.
Let$\phi \in W^{1,2}(D,X)$ and define
$$
W^{1,2}_{\phi}=\{v \in W^{1,2}(D,X):Trace(v)=Trace(\phi)\}
$$
Let $$
...

**2**

votes

**0**answers

77 views

### Laplacian on manifolds with corners

So far I've studied smooth riemannian manifolds and the Laplace-Beltrami operator associated to it. Therefore I know basic theorems like Stoke's theorem, divergence theorem, green's identities etc. ...