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.

learn more… | top users | synonyms

3
votes
2answers
90 views

No normal coordinates on general Finsler manifolds

I recently read a footnote in Chern's article stating that a non-Riemmanian Finsler manifold does not possess normal coordinates. As I'm still new to non-Riemmanian Finsler geometry I don't see why ...
11
votes
1answer
384 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 ...
4
votes
2answers
542 views

Applications of Gauss-Bonnet theorem

In wikipedia,I was pretty amazed to find a proof of fundamental theorem of algebra using Gauss Bonnet theorem. I think given how central it is to mathematics with its far reaching generalizations ...
3
votes
1answer
88 views

Carre du Champ, Subunit Paths and CC-metrics

Let the operator $L$ be given by $Lf(x):=\nabla\cdot (A\nabla f(x))$, where $f:\mathbb{R}^d\rightarrow \mathbb{R}$ belongs to a suitable class of functions $\mathcal{A}$. The carre du champ operator ...
4
votes
1answer
177 views

Is there an example of a Killing vector field on a complete Riemannian manifold with finite volume?

Is there a Killing vector field on a complete Riemannian manifold $M$ with finite volume that satisfies the condition $$\displaystyle\liminf_{r\rightarrow +\infty} \displaystyle\frac{1}{r} ...
6
votes
0answers
68 views

Curvature Characterization of Homogeneous Spaces

A (Riemnnian) symmetric space is locally characterized by the first covariant derivative of its curvature tensor, namely, a manifold $M$ is locally a symmetric space if and only if $\nabla R=0$ (where ...
3
votes
1answer
77 views

Parallel Transport on Hypersurface Spinor Bundle

So this has been driving me up a wall. I'm trying to digest parts of the Parker & Taubes paper, "On Witten's Proof of the Positive Energy Theorem." Here's a link: ...
0
votes
0answers
59 views

Weak $H^1$-limit of “almost conformal” maps

Let us consider a sequence of maps $\phi_n : (M, g) \to \mathbb{R}^k$, where $M$ is a surface. Let $\phi_n \to \phi$ weakly in the sense of $H^1$-norm, and let $\phi$ be non-trivial. Let $\mathcal{H}$ ...
4
votes
2answers
179 views

Heat kernel asymptotics for small distances

I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies $$p_t(x, y) = \frac{1}{(4\pi ...
-3
votes
0answers
71 views

Harmonic map into $S^n \times \mathbb{R}$ [closed]

Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) ...
0
votes
1answer
350 views

Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me: Is there any open Ricci-flat ALE 4-manifold other than ...
1
vote
0answers
63 views

Euclidean metric in spherical coordinates [closed]

I apologize if this question is not research level, but it has been asked on MSE before, and not received really satisfactory answers, for example, see here, and googling does not reveal anything ...
6
votes
2answers
238 views

Hodge map and the Cohomology Ring of a Riemannian Manifold

For a compact Riemannian manifold $M$, we know that the Hodge map $\ast$ and Laplacian $\Delta$ commute. From Hodge decomposition and its implied isomorphism between harmonic forms and cohomology ...
4
votes
0answers
193 views

stochastic control / geometric mean

Consider the following problem: Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...
3
votes
1answer
169 views

Totally geodesic subgroups in Lie groups

Let $G$ be a Lie group with a left invariant metric $g$. Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is right-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every ...
0
votes
0answers
29 views

Obtaining Hessian of the embedding from an induced metric

Consider a hypersurface (not necessarily compact) smoothly embedded into $\mathbb{R}^n$ such that the Hessian is a positive definite bilinear form. Due to positivenes, Hessian can be taken as a metric ...
5
votes
0answers
66 views

Regularity of the distance from the boundary in singular riemannian manifolds

I am looking for references related with the regularity of the distance from the boundary in singular Riemannian manifolds. I assume the following setting. $(M,g)$ is a Riemannian manifold, with ...
2
votes
1answer
141 views

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, ...
3
votes
0answers
71 views

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 ...
11
votes
1answer
208 views

Hausdorff convergence of submanifolds in Riemannian manifolds

Let $(M^n,g)$ be a smooth compact Riemannian manifold. It is well known that any sequence $\{X_i\}$ of compact subsets of $M$ has a subsequence which converges in the Hausdorff metric to a compact ...
4
votes
2answers
200 views

Regarding Ricci curvature of Markov chains

In Ricci curvature of Markov chains on metric spaces Yann Ollivier, defines a coarse Ricci curvature for a Markov chain with transition kernels $\{m_x\}$ defined on a metric space $(X,d)$ as follows: ...
3
votes
2answers
171 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?
1
vote
0answers
41 views

Understanding the Exp map from a moduli of smooth curves

The setup: Suppose I'm given a family of class $C^k$ curves $\{f_{z}\}_{z\in Z}$ with values in $\mathbb{R}$ parameter by a finite number of parameters $Z\subseteq \mathbb{R}^m$. Let $\mathscr{M}$ ...
1
vote
1answer
41 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 ...
4
votes
1answer
75 views

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 ...
3
votes
0answers
83 views

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 $T^1S^n$, ...
8
votes
1answer
116 views

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 ...
0
votes
0answers
92 views

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 ...
2
votes
1answer
60 views

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 ...
0
votes
0answers
88 views

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 ...
8
votes
2answers
388 views

Is displacement controled by stable norm?

Let $T^n$ be the $n$-dimensional torus and $g$ be a Riemannian metric on $T^n$. Let $\tilde g$ be the induced metric on the universal covering; using suitable coordinates, $\tilde g$ is therefore a ...
10
votes
2answers
374 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 ...
1
vote
1answer
212 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 ...
-2
votes
0answers
71 views

Parallel bi-linear forms on $S^2$

Let $S^2$ denote the sphere endowed with its standard (curvature $k= 1$) metric $g$. How many linearly independent, parallel, symmetric bi-linear forms can we find on $S^2$ ? Clearly the metric $g$ ...
6
votes
1answer
159 views

Hodge de Rham operator and orientability

Let $(M,g)$ be a Riemannian manifold. One can consider the exterior algebra bundle $\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by $\Omega^k(M)$. One can consider ...
6
votes
0answers
133 views

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 ...
1
vote
0answers
66 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 ...
3
votes
0answers
56 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 ...
7
votes
1answer
150 views

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 ...
4
votes
0answers
103 views

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? ...
4
votes
0answers
82 views

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, ...
5
votes
5answers
2k 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 ...
7
votes
1answer
813 views

Isoperimetry and Poincaré Inequality

What are the known relations between isoperimetric and Poincaré inequalities on manifolds? For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
2
votes
0answers
249 views

The geometry of the holomorph of a Lie group

Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$) Is Hol$(G)$ always a Lie group? If the answer is yes our main questions: 1.For a left ...
5
votes
0answers
167 views

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 ...
1
vote
0answers
58 views

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 ...
12
votes
6answers
3k views

Tensor contraction and Covariant Derivative

What is the importance and intuition behind the the contraction operator on tensors (or the trace of a matrix, for that matter)? In addition, I see that one of the requirements for a covariant ...
6
votes
0answers
143 views

$2-$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 ...
2
votes
0answers
97 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 ...
6
votes
1answer
294 views

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 ...