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 Riemann manifolds with constant injectivity radius

I'm interested in compact Riemann manifolds which have that property that the injectivity radius at a point $p$ doesn't depend on $p$. Another way to put this is that the function $$p \mapsto d(p, ...
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Flows associated with Killing fields

Let $M$ be a Riemann manifold and $p, q$ two points on a geodesic $\sigma$ which are isotropically conjugate. That is, there is a Jacobi field along $\sigma$ vanishing at $p$ and $q$ which is the ...
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185 views

Upper bound for Willmore energy

Good day to everyone! Does anybody know if there are upper bound estimates for Willmore energy for a given surface?
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54 views

material derivative and relation to Riemannian metric

For each $n$ let $N_t$ be an embedded smooth hypersurface in $\mathbb{R}^n$ of dimension $n-1$. $\{N_t\}_t$ is a family of hypersurface that is evolving with some velocity $V$. Smooth functions on ...
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Differentiability of a map to the free loop space

While reading Morse theory, closed geodesics, and the homology of free loop spaces, the author claims the following: Given the $S^{n-1} \hookrightarrow Y_1 \rightarrow T^1S^n$ bundle over $T^1S^n$, ...
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Uniform partitions of a compact Riemannian manifold

My question is a little bit vague. I want to know if an arbitrary compact Riemannian manifold (M^d,g) admits partitions that are uniform in some sense. To be more precise, I need for every eps > 0 a ...
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Stability for open manifolds of finite volume under lower Ricci curvature bound

By a theorem of Cheeger-Colding if $N_i\to M$, where $\to$ means Gromov-Haussdorff convergence, $N_i$ and $M$ are of the same dimension, compact and smooth Riemannian manifold with finite volume, as ...
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Constant Harmonic Mean surfaces

For surfaces embedded in $\mathbb R^3$ with principal curvatures $ \kappa_1, \kappa_2 $ we know bending/isometric mappings conserve $ K= \kappa_1 \kappa_2 $ and CMC DeLaunay type minimal surfaces ...
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Conformally flat with zero scalar curvature 2 [duplicate]

First, i am the one who asked about the existence of compact manifolds of dimension $n\geq 4$ which are conformally flat, non-flat, with zero scalar. Due to the fact that i couldn't comment because i ...
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395 views

A Converse to Cartan–Hadamard theorem?

Let $M$ be a complete Riemannian manifold, with the property that $\exp_p\colon T_pM \to M$ is a diffeomorphism for every $p \in M$. Can we say something about it's curvature? Is it true that its ...
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233 views

Soliton equation and non-killing potential vector field

I am searching for a non-Killing vector field $\zeta \in\frak X\rm (M)$ where $(M,g)$ is a Riemannian manifold such that $$\frac12 \frak L_\zeta \rm g+Ric=\lambda g$$ $$ \frak L_\zeta \rm Ric=\lambda ...
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Is there any exotic smooth structure on open hyperbolic manifold?

I edited my post to clarify some confusions as suggested by Igor. Let $M$ be an open hyperbolic manifold, with or without finite volume, Is there any manifold $N$ which is homeomorphic to $M$ but ...
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79 views

Gaussian heat kernel bounds on Riemannian manifolds

I wish to know if we have Gaussian lower and upper bounds for the heat kernel,i.e. $$ t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_1t}} \lesssim p_t(x,y) \lesssim t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_2t}}, $$ on a ...
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61 views

Minimal volume of manifold homotopic to a hyperbolic manifold with finite volume

Let $(X, g_0)$ be a $n$-dimensional open manifold with finite volume hyperbolic metric. Suppose $(Y, g)$ is another $n$-dimensional manifold and $f: Y\to X$ a proper degree one map. Then by Storm's ...
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Does $S^n\times H^k$ have non-isometric conformal transformations?

Question: Consider the product Riemannian manifold $(M,g)=(S^n\times H^k,g_{round}\oplus g_{hyp})$ of a round sphere $(S^n,g_{round})$ of curvature $1$ and hyperbolic space $(H^k,g_{hyp})$ of ...
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Geometric Characterization of Martingales

Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as Geodesics in a very large dimensional manifold. My question is, is there any research studying this idea? ...
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The scope of correspondence principle in quantum chaos

My understanding of the so-called correspondence principle in quantum chaos, is that it is a connection between the behaviour of a classical Hamiltonian system (chaotic/completely integrable) and the ...
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Is positively curved Alexandrov surface isometrically embeddable in $\mathbb R^3$?

I guess it is not. The example I have in mind is: $X^2$ is the spherical suspension of a circle $S^1(t)$ of length $0<t<2\pi$. Then $X$ has constant curvature =1 except at two suspension points, ...
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Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?

I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...
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Isotropically conjugate points in reductive spaces

Is it the case that a naturally reductive Riemannian homogeneous space that contains a pair of conjugate points necessarily contains a pair of isotropically conjugate points? That is, there is a pair ...
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107 views

Estimate for the first eigenvalue of the Laplacian

I was studying the paper of S. T. Yau - Seminar on Differential Geometry - and there asks if the first eigenvalue is equal to $ n $, if we have a embedded oriented Riemannian manifold and closed ...
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Can a complete manifold have an uncountable number of ends?

Let $M$ be a complete and noncompact Riemannian manifold. Fix a point $p$ in $M$. Let $\gamma$: $[0, L]\rightarrow M$ (parametrized by its arc length) be a geodesic starting from $p$. Denote by ...
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2answers
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Abelian isometry groups of codimension one

Good day. Let (M,g) be an n-dimensional Riemannian manifold (complete, if you wish), and suppose that there exists an n-1 dimensional Abelian group acting by isometries on M. Or locally, near a point ...
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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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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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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 ...
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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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214 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 ...
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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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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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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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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 ...
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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 ...
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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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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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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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 ...
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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 ...