Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.

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Compact Eucledean hypersurfaces with “almost” constant H_k curvature

Let $M$ be an Eucledean $n$-dimensional compact hypersurface with constant $H_k$ curvature, where $k=1,...n$. A theorem by A.Ros tell us that so $M$ is an Eucledean sphere. Does anybody know if there ...
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2answers
302 views

Volume form on a hyperbolic manifold with geodesic boundary

Let $M$ be a compact connected orientable Riemannian $n$-manifold with boundary $\partial M\ne\emptyset$. Since $H^n(M,\mathbb R)=0$, the connecting morphism $\delta: H^{n-1}(\partial M,\mathbb R)\to ...
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146 views

Question about the normal bundle of a totally geodesic submanifold of a Cartan-Hadamard manifold

I'm reading the exposition in Lang's Fundamentals of Differential Geometry of the following generalization of the Cartan-Hadamard theorem: Suppose $X$ is a Cartan-Hadamard manifold (i.e. a complete, ...
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$2-$conformal vector fields on Riemannian manifolds

A vector field $\zeta$ is conformal on a Riemannian manifold $(M,g)$ if $$\mathcal L_\zeta g=\rho g$$These vector fields have a well known geometrical interpretation. The flow of a conformal vector ...
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Is there metric of nonnegative sectional curvature with exact form on $TM$

Consider natural metric on $TM$ That is $$ g(X^h,Y^h) = g(X,Y),\ g(X^h,Y^v)=0$$ where $h,\ v$ mean horizontal and vertical lifts. There exist two well known metric of this type : Sasaki metric $g(X^v,...
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37 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 ...
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2answers
217 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
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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 ...
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1answer
59 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 ...
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147 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, ...
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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 ...
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1answer
143 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
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1answer
98 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 f(x,y)}{\...
7
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1answer
205 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 ...
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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 ?
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123 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
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1answer
120 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
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2answers
186 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 ...
2
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3answers
216 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 ...
3
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96 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 $f_{\...
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0answers
47 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 ...
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1answer
332 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 X=...
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1answer
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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 ...
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1answer
92 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 ...
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2answers
266 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 example,...
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1answer
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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 ...
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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 $$d([q],[q'])=\frac{\sup_{x\...
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2answers
269 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
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1answer
116 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
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1answer
210 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 \mathbb{...
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0answers
56 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 ...
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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
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1answer
102 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 ...
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1answer
366 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 ...
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117 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
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1answer
408 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 ...
9
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3answers
390 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 non-...
2
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2answers
335 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 ...
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5answers
518 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
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1answer
92 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
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2answers
150 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 M:\,\lim_{t\to-\infty}\...
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48 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 ...
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83 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, ...
5
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1answer
227 views

Convergence of Riemannian metrics spectra

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 ...
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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 ...
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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 $g$ ...
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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 $\...
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100 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 ...
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1answer
177 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 $P=\mathbb{...
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1answer
131 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 ...