**2**

votes

**1**answer

206 views

### The points of half area of a triangle

Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is ...

**0**

votes

**0**answers

47 views

### Quadric functions on a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold such that the parallel transport along every simple closed curve is the identity operator. A smooth function $f:M\to \mathbb{R}$ is called a quadric function if ...

**2**

votes

**1**answer

143 views

### totally geodesic submanifold of Heisenberg group

Let $G= \left\{ \begin{pmatrix} 1&a&c\\0&1&b\\0&0&0 \end{pmatrix} \mid a,b,c\in \mathbb{R} \right\}$ be the Heisenberg group. Is there a compact codimension one submanifold ...

**4**

votes

**1**answer

209 views

### What is the difference between $\delta W^{\pm}=0$ and Einstein?

Maybe this is a vague question. In Besse's book Einstein manifolds, $\delta W^{\pm}=0$ is considered as a generalization of Einstein metrics on four-manifolds. I was wondering what is the difference ...

**9**

votes

**1**answer

253 views

### Decomposition of $\mathrm{O}(n)$-modules coming from differential geometry

Let $V$ be a $n$-dimensional real vector space equipped with a positively definite scalar product $g$ and let $\mathrm{O}(n)$ be the automorphism group of $(V,g)$. View $V^{\otimes k}$ as a ...

**4**

votes

**1**answer

186 views

### Analytic representatives for Kahler classes

If we are given compact complex manifold $X$ and a Kahler class $[\omega]$,
can we always find a positive definite representative $\omega \in [\omega]$ that is
real analytic?

**2**

votes

**0**answers

421 views

### The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...

**3**

votes

**0**answers

62 views

### Hessian eigenspaces form integrable distributions on a Riemannian manifold?

Suppose $M$ is a Riemannian manifold and $f:M\to\mathbb{R}$ a differentiable function. I can form the Hessian $H$ of $f$ (with respect to the Levi-Civita connection); this is a symmetric bilinear ...

**3**

votes

**0**answers

120 views

### On the volume entropy of negatively curved manifolds

Let $X$ be the universal cover of a closed negatively curved Riemannian manifold. Let $x_0\in X$ be a base point, $S$ be the unit sphere in $T_{x_0}X$ and $\exp:T_{x_0}X\rightarrow X$ be the ...

**4**

votes

**3**answers

268 views

### Is the group of isometries of a homogeneous Riemannian manifold maximal?

I have a homogeneous Riemannian manifold X with isometry group Iso. Is Iso a maximal group? By maximal group, I mean that there does not exist another group G such that:
Iso is a proper subgroup of ...

**2**

votes

**0**answers

109 views

### Variational inequality on Manifold

Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla ...

**1**

vote

**3**answers

159 views

### Harmonic Function with special property

I would appreciate any help with the following problem:
Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two ...

**3**

votes

**1**answer

276 views

### A question on Schrodinger operator

I am not sure whether I should ask for help here or math stackexchange. I got trouble with an inequality involving the Schrodinger operator on manifolds. Any suggestion is appreciated!
Let $(M,g)$ be ...

**1**

vote

**1**answer

124 views

### Volume bounds of balls in Riemannian manifolds

Let $(M,g)$ be a complete Riemannian manifold and suppose $\mathrm{Ric}(g) \geq -k$ for some $k>0$. Suppose we know that $\mathrm{vol}_g (B_1^g (x_0)) \geq \nu$ for some particular $x_0 \in M$ and ...

**5**

votes

**1**answer

298 views

### Volume of geodesic balls

I have two questions (somewhat related) regarding local geometry on a SMOOTH, COMPACT Riemannian manifold. I still have a hard time getting a "good" understanding of local geometry.
Question 1:
It ...

**3**

votes

**0**answers

133 views

### Counterexample to volume comparison inequality assuming only scalar curvature bound?

The Gromov-Bishop volume comparison theorem says that if we have a lower bound for the Ricci curvature on $(M,g)$, then its geodesic ball has volume not greater than the geodesic ball with the same ...

**1**

vote

**0**answers

63 views

### Coarse geometry of minimal surfaces in non-positively curved manifolds

Let $X$ be a simply-connected Riemannian manifold of non-positive curvature and $S\subset X$ be a complete minimal surface.
(You can basically image $X$ as a ball and $S$ as an embedded disk whose ...

**3**

votes

**0**answers

116 views

### $\mathbb{CP}^1$-structures and hyperbolic Gauss maps

Let $\Sigma$ be a closed surface of genus at least $2$.
Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...

**-5**

votes

**1**answer

135 views

### compact complex manifolds and complet curves [closed]

let $X$ be a compact complex manifold of dimension one .
my first question is : 1) -does all compact complex manifolds of dimension one admit
nonconstant meromorphic function ? .
now , let ...

**3**

votes

**1**answer

339 views

### Taylor expansion of the determinant of a Riemannian metric

Let $(M,g)$ be a compact Riemannian manifold without boundary. Fix a point $x\in M$ and $N\ge 2$ large. Then there exists a metric $\tilde g$, conformal to $g$ such that $$ \det \tilde g=1+O(r^N)$$ ...

**1**

vote

**0**answers

89 views

### A question on the maximum principle for Schrodinger operators

Let $(M^n,g)$ be a closed Riemannian manifold, and $L=-\Delta+V$ be a Schrodinger operator, $V\in C^{\infty}(M)$. In answers to the two questions (First eigenvalue of Schrödinger operator is simple 1 ...

**1**

vote

**0**answers

120 views

### Estimate the smallest eigenvalue of a Schrodinger operator

There are several results on the estimate of the number of negative eigenvalues of a Schrodinger operator, see a recent paper of Grigor'yan-Nadirashvili-Sire and references therein. I wonder how to ...

**0**

votes

**1**answer

197 views

### Describe all differentiable functions on $\mathbb{S}^n \backslash S$ (S is the south pole) [closed]

Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely ...

**2**

votes

**1**answer

132 views

### local approximation of a vector field on a Riemannian manifold

Let $(M^n,g)$ be a Riemannian manifold, and let $V$ be a $C^{\infty}$ vector field on $M$. Is it possible to locally approximate $V$ by gradient vector fields $\nabla f_i$, such that the ...

**0**

votes

**0**answers

70 views

### Local behavior of Killing spinor on Sasaki-Einstein Manifold

I am trying to understand how a Killing spinor behaves near a closed Reeb orbit, for instance, on $S^5$ and $Y_{p,q}$ manifolds
So Let us consider the Killing spinor equation on a five-dimensional ...

**2**

votes

**2**answers

329 views

### Sharp Gaussian upper bounds on Heat Kernel

I am looking for references (with proof) for the following statement:
Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be ...

**1**

vote

**0**answers

127 views

### Application of conformal normal coordinates for higher order elliptic operator

Let $n>2$ be even. Consider a compact Riemannian manifold $(M^n,g)$ and denote with $P_g$ the critical GJMS operator.
Recall that $P_g$ is conformally invariant, i.e.
$$P_{\tilde g}=e^{-nu}P_g$$ ...

**1**

vote

**1**answer

160 views

### de Rahm Laplace operator on forms bounded

Let $M$ be a closed differentiable manifold. Let $E^{p}(M)$ be the vector space of $p$-forms on $M$ equipped with the $L^{2}$-inner product $(\alpha, \beta) = \int_{M}\alpha \wedge \star \beta$. The ...

**3**

votes

**0**answers

171 views

### Can we obtain topology results using analysis in metric measures spaces?

Let $M$ be a smooth compact manifold. It is known that a lower bound on the Ricci curvature is equivalent to the convexity of the entropy on $\mathcal{P}^2(M)$ (Von Rennesse and Sturm '05), but I ...

**1**

vote

**2**answers

407 views

### Geodesic equation from Christoffel symbols

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the manifold of all (strictly positive) probability vectors (distributions) on $\mathcal{X}=\{x_0,\dots,x_n\}$,
i.e., each $p=(p(x_0),\dots,p(x_n))\in ...

**1**

vote

**1**answer

133 views

### Is the structure constant additive on connected components?

Let $M$ be a Riemann surface and $\mu$ a metric on it, which could be non-compact. Moreover let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and $\mathrm{det}^*(\Delta_{\mu,\,M})$ its ...

**0**

votes

**0**answers

76 views

### Decomposition of the canonical flat connection on $\tilde M\times_{\rho} SL(n,\mathbb{C})$

I'm looking for a proof resp. reference for a statement of the following form:
Let $M$ be a compact Riemann surface, $\tilde M$ its universal covering, $\rho$ a semisimple representation of its ...

**1**

vote

**0**answers

57 views

### Bounds on functions pullbacked via exponential map

Let us assume that $M$ is a compact Riemannian manifold (without boundary). For any point $x\in M$, we can pullback $C^\infty(M)$ functions to $T_x M$ via the exponential map, by setting
$$ (\exp_x^* ...

**0**

votes

**1**answer

76 views

### Optimal Regularity for Invariance of Curvature under Isometries

It is well known that sectional curvature is an invariant under isometries. I wonder what the optimal regularity for this result to hold is (in terms of Hölder-spaces)?. It is classical that ...

**12**

votes

**2**answers

936 views

### Does there exist a closed manifold that can be given both a Euclidean and a Hyperbolic structure?

I originally asked this on math.stackexchange, where I asked if there could exist a closed manifold that could be given different geometric structures of constant curvature (not at the same time, of ...

**3**

votes

**1**answer

253 views

### Special Riemannian connections?

Assume that $E$ is a bundle of Lie Algebras. Let $g$ be an invariant metric on $E$, that is for all $p\in M$, $$g_p([x,y],z)+g_p(y,[x,z])=0,$$ where $x,y,z\in E_p$ are arbitrary.
Is there a ...

**11**

votes

**2**answers

597 views

### Obtain Lorentzian manifolds from Riemannian ones by Wick rotation

In some cases, Wick rotation of a metric, formally consisting in substituting a coordinate with i times the coordinate itself, allows one to construct a Riemannian manifold starting from a Lorentzian ...

**2**

votes

**0**answers

135 views

### Notions of convexity for the boundary of a Riemannian manifold

Let $(M,g)$ be a compact smooth Riemannian manifold with boundary $\partial M\subset M$.
Question: What does it mean to say that the boundary is convex and strictly convex?
Alternatively: What ...

**1**

vote

**1**answer

152 views

### Relation between volume of reduced space and phase space

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is ...

**5**

votes

**1**answer

128 views

### Regularity of Patterson-Sullivan Length function

Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function,
...

**1**

vote

**1**answer

208 views

### A special type of transitivity

Let $M$ be a smooth orientable manifold with volume form $\Omega$. Fix two pints $x,y \in M$. Put $A$=all volume preserving diffeomorphism of M which maps $x$ to $y$.
Define $B$=All linear volume ...

**10**

votes

**2**answers

322 views

### A Scalar Curvature Computation in Brendle Marques Neves' Min-Oo Conjecture paper

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition:
Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...

**5**

votes

**1**answer

173 views

### Question about conjugate points

If there exist two geodesics from $p$ to $q$ that are not only different from each other but also infinitesimally close to each other, then it implies that $q$ is conjugate to $p$.
Can anyone give an ...

**2**

votes

**1**answer

238 views

### If there exists a nontrivial vector field $V$ such that $\nabla_{X}V=0$ for any vector field $X$, the manifold must be flat?

If there exists a nontrivial vector field $V\not=0$ in Riemannian manifold $M$ and an open set $U\subset M$ such that $\nabla_{X}V=0$ in $U$ for any vector field $X$ in $M$, then dose $U$ have to be ...

**1**

vote

**1**answer

101 views

### Factor of 2 In the Definition of Metric Contact Structure

In Blair's book and many many literatures, I see definition of a contact metric manifold which involves a relation
\begin{equation}
d\kappa \left( {X,Y} \right) = g\left( {X,\Phi Y} \right)
...

**8**

votes

**3**answers

415 views

### How to define the square root of $1-\Delta $?

If $M$ is a Riemannian manifold with $\Delta $ its Laplacian, how can we define $(1-\Delta)^{1/2}$?
The book I am reading says that $(1-\Delta)^{1/2}$ is an invertible first-order pseudo-differential ...

**1**

vote

**1**answer

150 views

### Riemannian metric and Volume form for $SE(n)$ and/or $E(n)$

I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the (left ...

**5**

votes

**5**answers

1k views

### Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?

Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat?
I am especially interset in the case ...

**-2**

votes

**1**answer

143 views

### Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)

Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question
We consider the following two classes of smooth maps on ...

**1**

vote

**1**answer

124 views

### Connection on canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on symplectic manifold

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic 4-manifold $(M, \omega)$, with a compatible $J$ and $g$, so
\begin{equation}
{W_ + } = {T^{0,0}}{M^*} ...