# Tagged Questions

273 views

### Behavior of the spectrum of the Laplacian under pointed smooth convergence

The Laplacian on a compact Riemannian manifold has a discrete spectrum. For example on a circle of perimeter $L$ the $n$-th eigenvalue starting at $0$ is $-\lambda_n = -(2\pi/L)^2 n^2$. On the other ...
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### 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 ...
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### Infinite dimensional Riemannian geometry

My current research has brought me into an area the requires me to learn some infinite dimensional Riemannian and Kähler geometry. Can someone recommend some good books or survey articles to help me ...
392 views

### A question on generalized Einstein metrics on four-dimensional manifolds

I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds, \begin{equation*} ...
361 views

### Ricci curvature under rough convergence

From the work of Lott--Villani and Sturm, I know that the following fact holds: (*) Suppose that $(M_k,g_k,dvol_{g_k})$ is a sequence of compact Riemannian manifolds of non-negative Ricci ...
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### Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
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### Pullback of $L^p$ functions via exponential map

Let $M$ be a complete Riemannian manifold, endowed with its exponential map $\exp: TM \longrightarrow M$. For any $C^k$- function $u$, we get the Pullback $$\exp^* u = u \circ \exp$$ which is in ...
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### Monograph or rich survey on infinite-dimensional Riemann manifolds

I'm working with the space of smooth curves $\mathcal{C}$ in a smooth manifold $M$, having (different, pre-determined) fixed endpoints. I'd like to endow it with a Riemann structure (I already have a ...
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### Reference request for an early theorem of Gromov

In his talk Misha Gromov- How does he do it, Jeff Cheeger mentions a theorem of Gromov proved sometime in the early 70's. Theorem: Every manifold admitting a sequence of metrics such that the diameter ...
534 views

### Reference request: Geodesic flow on a manifold with negative curvature is ergodic

I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result: The geodesic flow on a manifold with negative curvature is ergodic. The lecture note that ...
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### Spectrum of Laplacian in non-compact manifolds

What can be said about the spectrum of the Laplace-Beltrami operator on a non-compact, complete Riemannian manifold of finite volume? For example, is the point spectrum non-empty? What would be a ...
146 views

### Derivative of (the length of) the Ricci tensor

I was wondering, have you ever seen a formula in the Riemannian (more specially Kahlerian but not essential) setting for the derivative $X \cdot |Ric|^2 = 2 g(\nabla_X Ric, Ric)$ for a vector field ...
231 views

### Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?

On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...
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### Curve on a surface defined by its geodesic curvature

Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its ...
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### What is “Berger's isembolic inequality”?

Googled the name, but almost all result pointed to Berger's preprint. Is there any reference for this?
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### Reference request for parallel transport

I am learning about parallel transport on a Riemannian manifold equipped with an affine connexion. It seems (if I understand it well) that, in general, we might not be able to compute the parallel ...
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### Geometric meaning of a certain form in almost-Kähler geometry

I have difficulties finding an appropriate reference for the following question: Let $(M^{2n},g,J,\omega)$ be a compact almost Kähler manifold. Let $\operatorname{ric}$ the usual Ricci tensor of ...
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### Voronoi cells and the dual complexes in Riemannian manifolds

I would like to use some "intuitively clear" properties of Voronoi cells in general Riemannian manifolds, but I have trouble finding references. Let $(X,d)$ be a connected Riemannian manifold and ...
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### Reference request: Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$

Does anyone know a citeable reference which works out the properties (geodesics, geodesic distance, ect) of the Riemannian manifold of linear isometries from $\mathbb{C}^n$ into $\mathbb{C}^m$, ...
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### How to generate a random (Weyl) curvature operator ?

Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity : ...
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### Fundamental groups of compact manifolds with non-negative Ricci curvature.

I would like to find an appropriate reference for the following statement: Statement. Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature. Then $\pi_1(M)$ is virtually abelian. ...
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### Open problems about CMC hypersurfaces with symmetries?

Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is ...
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### G-structures and complete riemannian manifolds

what are possible fundamental and introductory texts about G-structures ? and where i can find the proof of this proposition: if G(group) acts properly discontinuously on a space X , then G is a ...
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### Heisenberg group: research themes

I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties. I would like to know ...
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### How the Jacobi metrics may be useful in mechanics with or without constraints?

A mechanical system $(Q,K,V)$ is specified by the configuration space $Q,$ the potential energy $V\in C^\infty(Q),$ and the kinetic energy $K=K_g$ given by a Riemannian metric $g$ on $Q.$ If ...
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### Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature

A curve in the plane is determined, up to orientation-preserving Euclidean motions, by its curvature function, $\kappa(s)$. Here is one of my favorite examples, from Alfred Gray's book, Modern ...
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### Convex hull on a Riemannian manifold

Let $M$ be a complete Riemannian 2-manifold. Define a subset $C$ of $M$ to be convex if all shortest paths between any two points $x,y \in C$ are completely contained within $C$. For a finite set of ...
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### Is there more than one closed geodesic on $S^3$?

I know from two sources that it is (or at least was) unknown whether there are infinitely many geometrically distinct closed geodesics for every Riemannian metric on $S^3$, the 3-sphere (Weinberger, ...
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### Higher derivatives than Jacobi fields.

Hi, the first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the ...
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### Helmholtz-Decomposition on compact Riemannian manifolds

For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty$ into a "gradient"- and a "divergence-free"-part such that ...
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### convergence theory -> lorentzian geometry

Does someone have examples of extensions of results from convergence theory for riemannian geometry to a lorentzian setting. (I am familiar with the work of M.T.Anderson and co. in CMC gauge, i would ...
334 views

### Sasaki Metric of the Tangent Bundle over the Hyperbolic Plane

This is a reference request on what are surely well known facts. Let $M$ be a compact hyperbolic surface and $S(M)$ its unit tangent bundle. It follows from facts about Möebius tranformations in the ...
1k views

### The Origin of the Musical Isomorphisms

In Riemannian geometry, the "lowering indices" operator is denoted by $\flat:TM \to T^*M$ and the "raising indices" operator by $\sharp:T^*M \to TM$. These isomorphisms are sometimes referred to as ...
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### Basic results in bounded geometry

I'm doing analysis (dynamical systems) in the context of Riemannian manifolds of bounded geometry and I find myself reproving quite a few standard results/tools from standard differential geometry, ...
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### Cut Locus in a Graph

I am wondering if the concept of a cut locus has been defined and explored in discrete graphs, rather than their usual home on manifolds? The Wikipedia definition (which I believe I (co-?)authored) ...
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### Special Killing Vector Fields

Consider $(M^{n},g)$ to be a Riemannian manifold and suppose that $X$ is a smooth non-trivial Killing vector field on $M$. Away from the zeros of $X$ we have a natural distribution $D$ of ...
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### The geometry of Nadirashvili's complete, bounded, negative curvature surface

I would like to understand the geometric structure of a surface that Nadirashvili constructed which resolved what was known as Hadamard's Conjecture. Perhaps in the 15 years since his construction, ...
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### Algebraic surfaces and their (intrinsic) geometry

Recently I began to consider algebraic surfaces, that is, the zero set of a polynomial in 3 (or more variables). My algebraic geometry background is poor, and I'm more used to differential and ...
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### Harmonic coordinates on Riemannian manifolds

I'm trying to read the paper of Jost and Karcher on the existence of harmonic coordinates on a ball whose size only depend on the injectivity radius and a two sided bound on the curvature. ...
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### What spaces have well known horofunctions?

Following Gromov, take a metric space $(X,d)$ and consider $C(X)/\mathbb{R}$ the set of continuous functions to $\mathbb{R}$ with the topology of uniform convergence on compact sets after taking the ...
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### Roadmap to learning about Ricci Flow?

Hello, I'm curious to what books etc. one could use to understand the basics of Ricci flow, what areas of math are needed and so? What areas should one specialize in? See it as a roadmap to ...
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### How does curvature change under perturbations of a Riemannian metric?

Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then they should have ...
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### Linear/Non-linear sigma model

This is slightly an open-ended invitation to discuss references and reasons for excitement about the linear and non-linear sigma model. I gauge from some other interactions that it has considerable ...
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### A Riemannian metric on S^2 \times S^2 of nonnegative curvature that is not a product

Good afternoon, there is an example of a Riemannian metric on S^2 \times S^2 of nonnegative sectional curvature that is not a product metric. I know there is one; however, I cannot find a specific ...
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### Contracting a geodesic on a space of curvature less than 1

I would like to ask for a reference to the following statement (hopefully correct): Let $M$ be a manifold of sectional curvature at most $1$ and let $\gamma$ be a closed geodesic. Suppose that ...
This question is edited following the comment of Joseph. He pointed out that the main object of the first vesrion of this question is the cut locus. Recall that the cut locus of a set $S$ in a ...
Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE \ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...