**0**

votes

**0**answers

34 views

### On a conjecture about Riemannian metric with positive sectional curvature [duplicate]

What is the last status of the following conjecture? Is it still open? What partial or similar results are known up to now?
Conjecture: $S^2\times S^2$ admits a Riemannian metric with positive ...

**10**

votes

**2**answers

176 views

### Spectral properties of the Laplace operator and topological properties

Suppose that $M$ is a closed Riemannian manifold: one can construct the so called Laplace-Beltrami operator on $M$. Its spectrum contains some information of the underlying manifold: for example its ...

**6**

votes

**1**answer

219 views

### What is known about Lie groups with positive(strictly) curvature?

If we consider $G$ a Lie group with left invariant riemannian metric its sectional curvature is nonnegative, when this metric is positive?
I thought a little about and only found $SU(2)=S³$.
In ...

**1**

vote

**1**answer

52 views

### Spaces of Killing spinors for different orientation

Simply put, I want to understand how a change of orientation on a Riemannian spin manifold can change the space of Killing spinors.
To be more precise:
Let $M$ be a spin manifold (i.e. the first and ...

**2**

votes

**0**answers

122 views

### Structure of $C^k$ ($k<\infty$)Riemannian metrics on a manifold

$M$ is a smooth manifold. It's known that if $M$ is compact, then the space of smooth Riemannian metrics has a Frechet manifold structure. For the space of $C^k$($k<\infty$) Riemannian metrics, ...

**4**

votes

**0**answers

143 views

### Obtaining the metric from the mixed Ricci tensor $R^i{}_j$

In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$).
But what do we know about ...

**3**

votes

**1**answer

131 views

### A clarification regarding analytic perturbation of metrics and Laplacian

This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...

**2**

votes

**1**answer

94 views

### Prescribing an induced metric

We know that, if we have a surface $z=f(x,y)$ with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form):
$$g=\begin{bmatrix}
1+\left ( \frac{\partial ...

**7**

votes

**1**answer

201 views

### Frucht's type theorem for Riemann surface

Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite ...

**4**

votes

**0**answers

102 views

### classification of homogenous complex manifolds

Suppose $X$ is a complex manifold (doesn't assume it's Kahler), and it's holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ?

**0**

votes

**0**answers

105 views

### Conformal and Killing vector fields

This is a continuation of my previous question that due to quid's advice I posted as a separated question.
We know that there is a close similarity between conformal and Killing vector fields on a ...

**7**

votes

**1**answer

104 views

### When a Riemannian manifold with boundary is an Alexandrov space?

Let $M$ be a smooth Riemannian manifold (without boundary). Let $X\subset M$ be a smooth compact submanifold with boundary, $\dim X=\dim M$.
Under what conditions $X$, equipped with the induced ...

**4**

votes

**2**answers

175 views

### Compact surface with arbitrarily large eigenvalue

Consider a compact surface $M$ with genus $\gamma \geq 2$ and fix a positive real number $V$. Is it known whether it is possible to produce a metric $g$ on the surface $M$ such that $(M. g)$ has ...

**1**

vote

**3**answers

201 views

### 1-parameter group of a vector field

Let $(M,g)$ be a Riemannian manifold and $\nabla$ be the Levi-Civita connection of $g$ and let $X,Y$ be vector fields on $M$. If $\lbrace \phi _t \rbrace $ is the 1-parameter group of $X$ then what is ...

**2**

votes

**0**answers

71 views

### Relation between harmonic vector fields and harmonic maps

Let $f:M\longrightarrow N$be a smooth map between Riemannian manifolds and $X\in \chi (M) $ be a harmonic vector field.
What are some necessary and sufficient conditions for guaranting that ...

**0**

votes

**0**answers

46 views

### Christoffel symbols on a loop group in Riemann normal coordinates

Christoffel symbols on a Lie group in Riemann normal coordinates
My question is a generalization of the question in the link above. How does one find the explicit form of the Christoffel symbols and ...

**1**

vote

**1**answer

306 views

### On Harmonic Unit Vector Fields

When we restrict the Dirichlet energy functional to the set of all unit vector fields on a compact Riemannian manifold $(M,g)$, then the critical points of this functional are satisfied in $\Delta_g ...

**4**

votes

**1**answer

164 views

### Is this distribution completely non integrable?

We consider the usual Riemannian metric on $S^{n}$. Its corresponding LC connection gives us a distribution on $TS^{n}$. Is this distribution completely nonintegrable?
In general, what type of ...

**1**

vote

**1**answer

89 views

### Lamination as limit of arcs

I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...

**5**

votes

**2**answers

254 views

### Which surfaces admit unbounded-length simple geodesics?

Let $S$ be a surface embedded in $\mathbb{R}^3$.
A simple geodesic on $S$ is one that does not self-intersect.
Some surfaces have simple geodesics whose length exceeds any
given bound $L$. For ...

**11**

votes

**1**answer

171 views

### Is there a proof of the uniformization theorem using circle packing?

In this paper: http://www.dm.unipi.it/~benedett/rodin-sullivan.pdf
Rodin and Sullivan show that circle packings converge to the Riemann map. Later, Scharmm and He found another proof of the same ...

**3**

votes

**0**answers

157 views

### Distance between quadratic forms

In notes here http://math.univ-lyon1.fr/homes-www/gille/prenotes/lens.pdf on page $2$ a formulation of distance between two positive quadratic form $[q],[q']$ is given by
...

**5**

votes

**2**answers

206 views

### Is every open convex subset of a Riemannian manifold necessarily contractible?

Question: Is every open convex subset $C$ of a Riemannian manifold $M$, necessarily contractible?
Here by a "convex subset" I mean a set $C$ having the property that between each pair of points in ...

**5**

votes

**1**answer

105 views

### Geometry of convex subsets in Alexandrov space/ Riemannian manifold

Let $X^n$ be an $n$-dimensional complete Alexandrov space with curvature bounded below (or a smooth Riemannian manifold, possibly with boundary). Let $U\subset X$ be an open dense subset with the ...

**2**

votes

**1**answer

192 views

### Is it true that given any two point in $M$ if there exists an unique geodesic joining those two points, then $M \sim \mathbb{R^n}$ [closed]

This following doubt initially came to my mind while thinking the relationship between number of genus of a manifold and number of geodesic between given two points.
DOUBT: Suppose $M\subset ...

**1**

vote

**0**answers

48 views

### Change of curvature by parallel transport

If $c$ is a normal geodesic and if $e_1$ is a unit parallel vector field, then assume that for unit vector field $v,\ v\perp e_1$, $$ R(e_1,v,v,e_1)(t) \leq R(e_3,e_4,e_4,e_3)(t) \ \ast$$
for any ...

**1**

vote

**0**answers

32 views

### Transition functions under harmonic coordinate

Assume $M$ is a manifold. Assume $M$ is covered by domains $B_i$ and $\phi_i: B_i\to B_1(0)\subset{\mathbb R}^n$ are harmonic coordinates.
The Laplacian operator under a harmonic coordinate
has a ...

**3**

votes

**1**answer

96 views

### Reference: Finsler Derivative?

On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...

**8**

votes

**1**answer

322 views

### Multiplicity of Laplace eigenvalues

Disclaimer: This is a very heuristic question and I will be satisfied with heuristic insights, if rigorous and precise answers are not possible.
All the examples of closed surfaces (or higher ...

**2**

votes

**0**answers

109 views

### If G-invariant metric is always Kahler-Einstein

Suppose there is an Hermitian symmetric space of compact type X. It is realized in the following way:
$X\hookrightarrow\mathbb{P}^N$ and equip it with induced Fubini-Study metric g. What's more, the ...

**11**

votes

**1**answer

388 views

### Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?

$\newcommand{\til}{\tilde}$
Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds.
Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...

**8**

votes

**3**answers

361 views

### Isometry group of a compact hyperbolic surface

Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have ...

**2**

votes

**2**answers

313 views

### Ricci flow and isometry group

It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change ...

**9**

votes

**5**answers

479 views

### List of generic properties of Riemannian metrics

I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...

**3**

votes

**1**answer

84 views

### Nearly length minimizing paths are close to geodesics? [closed]

Let $M$ be a Riemannian manifold which is geodesically convex.
It's known that length minimizing curves are geodesics (after a possible reparametrization).
Now fix* points $p,q \in M$
Is the ...

**8**

votes

**2**answers

143 views

### Banach manifold of paths with endpoints on submanifolds

Fix a Riemannian metric on a manifold $M$. Suppose that we fix two points $x,y \in M$. We start with the space
$C^{\infty}_{\searrow}(x,y) = \left\{\gamma: \mathbb{R}\to ...

**3**

votes

**0**answers

43 views

### Laplacian Spectra on Nearly Nodal Riemann Surfaces

Consider a family of complex curves ${\mathcal C} \to {\mathbb D}$ such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of ...

**1**

vote

**0**answers

79 views

### Mean value operator on Riemannian manifold

Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$)
Consider the mean value operator, ...

**4**

votes

**0**answers

161 views

### Convergence of spectrum

Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...

**3**

votes

**0**answers

59 views

### Geodesic balls in warped product spaces

Let $g_S$ be a Riemannian metric on the $n$-dimensional sphere $S^{n}$ and consider the space $M=(0,a)\times S^{n}$ with the warped metric
$g=dt^2+f(t)^2g_S$, where $f\colon [0,a)\to \mathbb{R}$ is a ...

**4**

votes

**2**answers

162 views

### Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$?

On a compact Kahler manifold, let $g$ be the unique Kahler-Einstein metric with $\mathrm{Ricc}(g)=-g$, proved to exist by Yau and Aubin when the first Chern class $C_1(M)<0$.
Question: Does $g$ ...

**7**

votes

**0**answers

221 views

### A question on a result of Colin de Verdiere

Consider a compact connected surface $M$ of some genus $\gamma \geq 2$. A particular case of a famous result of Colin de Verdiere (see here) says that if we fix $\gamma$ and select a finite sequence ...

**3**

votes

**0**answers

94 views

### Faster (than normal) convergence of the normalized Ricci flow on surfaces

Consider a compact surface $M$ of genus $\gamma > 1$ (I am using the more usual letter "$g$" to denote metric), and the normalized Ricci flow on it. It is known that at time $t$, the scalar ...

**6**

votes

**1**answer

168 views

### Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$?

Let
$$
M:=\{(P,\pi)\mid P\not\in\pi\}\subset\mathbb{P}^n\times\mathbb{P}^{n\,\ast}
$$
be the open and dense (and as such $2n$-dimensional) subset of non-incident point-hyperplane pairs. If ...

**3**

votes

**0**answers

41 views

### Foliation by Umbilic surfaces

Suppose $(M,g)$ denotes a Riemannian manifold with boundary that is a foliation by Umbilic surfaces. (As an example consider a manifold where the exists a unit parallel vector field) .
Is it ...

**4**

votes

**1**answer

126 views

### Is the heat kernel for the hyperbolic plane uniformly continuous in $t\in(0,\infty)$?

let $\mathbb{H}$ be the hyperbolic plane and let $k(t,x,y)$ be the associated heat kernel.
I am wondering, if for any fixed $y\in M$ and $\epsilon >0$ the function $u_t(x):=k(t,x,y)$ is continuous ...

**5**

votes

**0**answers

78 views

### On Holonomy in (regular) Riemannian Foliations

Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid:
Let $\mathcal{F}\subset M$ be a ...

**3**

votes

**0**answers

78 views

### A tangential fixed point property for manifolds embedded in Euclidean spaces

Assume that $M$ is a compact orientable manifold which is embedded in some Euclidean space $\mathbb{R}^{N}$
We say that $M$ has the tangential fixed point property if for every continuous $f:M\to ...

**1**

vote

**0**answers

50 views

### Geodesically convex neighborhood in Finsler manifolds

It is well known that every point of a Riemannian manifold $(M,g)$ possesses a fundamental system $\{U_n\}_{n\in\mathbb N}$ of geodesically convex neighborhoods. This means that every pair of points ...

**9**

votes

**1**answer

203 views

### Conformal changes of metric and geodesics

Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...