Tagged Questions

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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2
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1answer
108 views

Compact Riemannian manifold with maximum average distance

Given a compact connected Riemannian $2$-manifold $M$ with positive curvature (thus by Gauß-Bonnet, $M$ is diffeomorphic to a 2-sphere) and diameter 1, what is the supremum (as $M$ varies over all ...
4
votes
2answers
200 views

Some facts about cut-locus

Let $M$ be a 2-dimensional closed Riemmanian manifold diffeomorphic to $S^2$. S.B.Myers says "the cut-locus of every point $x\in M$ is a finite tree." How the set of point can be a tree? ...
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0answers
73 views

Deduce global estimate from scaling-invariant local estimate

Let $(M,g)$ be a non-compact Riemannian manifold, with finite volume (or compactly exhausted, or any nice condition you would like, except for compactness). Suppose I have a tensor $T$ on $M$ of which ...
8
votes
2answers
617 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 ...
1
vote
1answer
133 views

lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
9
votes
3answers
428 views

Number of disjoint simple closed geodesics

According to Jairo comment on the first version of this question I revise the question as follows; Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that: There are at most a ...
2
votes
0answers
190 views

Surface with bounded torsion propery [closed]

Let $S$ be a surface in $\mathbb{R}^{3}$ with the following property: There is a uniform constant $M$ such that for every Frenet curve $\gamma(t)$, contained in $S$, $| \tau(t) | \leq M$, for all ...
2
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0answers
195 views

Is the tangent bundle of hyperbolic space trivial?

Let $H$ be hyperbolic n-space. Let $TH$ be the tangent bundle of $H$, endowed with its Sasaki metric. I have two questions: Is $TH$ isometric to $H$ times a flat n-space? What is the group of ...
0
votes
1answer
222 views

On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
9
votes
1answer
331 views

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 ...
7
votes
4answers
724 views

Smoothing of the distance function on a Riemannian manifold

Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let ...
2
votes
1answer
149 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 ...
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0answers
43 views

Rational homogenous functions

I'm interested in the set $\mathcal{S}$ of rational functions $F \colon \mathbb{R}^3 \to \mathbb{R}$ verifying: \begin{align} \Delta F=0 \quad \text{et} \quad F(\lambda x)= \lambda^d F(x) \quad d \in ...
6
votes
3answers
231 views

Is the set of focal points of a submanifold on a normal geodesic discrete?

Let $M$ be a complete riemannian manifold, $L$ a smooth submanifold of $M$ and $\gamma$ a geodesic with $\gamma'(0)$ normal to $L$. A focal point of $L$ is a critical value of the normal exponential ...
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0answers
413 views

Bi invariant Riemannian metric on a Lie Group

I'm trying to find an example of a Lie group $G$ which admits a bi-invariant Riemannian metric, and which has a closed subgroup $H$ such that the manifold $G/H$ does not admit a $G$-invariant ...
5
votes
2answers
167 views

Are ramified covering of negatively curved manifolds negatively curved?

Gromov and Thurston proved in "Pinching constants for hyperbolic manifolds" that any finite ramified covering of a compact hyperbolic manifold, along a codimension $2$ totally geodesic submanifold, ...
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0answers
60 views

Existence of harmonic maps between loops

Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy ...
2
votes
1answer
39 views

Prescribing finitely many unparameterised planar geodesics

Given a finite collection of embedded $C^\infty$ curves which pass through the origin in $\mathbb{R}^2$ with different tangent directions and never again intersect, is there a clean way of prescribing ...
-1
votes
2answers
175 views

On the definition of convergence of a sequence of sections of a bundle

Convergence of a sequence of sections of a bundle is defined as follows: Definition: Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on ...
5
votes
1answer
192 views

Can an open manifold with positive Ricci curvature be non simply connected at infinity?

The question is in the title, I haven't been able to locate a discussion of these kind of properties.
3
votes
1answer
237 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 $0$ ...
11
votes
3answers
254 views

Are there quanitative versions of Thurston's geometrization for manifolds which fiber over $S^1$?

The geometrization theorem tells us: Theorem (Thurston) The mapping torus $M_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S_g \rightarrow S_g$ from a genus $g$ surface to itself admits a ...
2
votes
1answer
219 views

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 ...
15
votes
1answer
270 views

Avoiding integers in the spectrum of the Laplacian of a Riemann surface

Let $\Sigma^2$ be an orientable compact surface of genus $gen(\Sigma)\geq2$, and denote by $\mathcal M(\Sigma)$ the moduli space of hyperbolic metrics on $\Sigma$, i.e., Riemannian metrics of constant ...
1
vote
1answer
114 views

What is “Berger's isembolic inequality”?

Googled the name, but almost all result pointed to Berger's preprint. Is there any reference for this?
1
vote
1answer
251 views

How to understand two examples of spin bundle

I am confused by two examples of spinor bundles over 4-manifolds, which I saw in various places: (1) The spinor bundle $S = S_+ \oplus S_-$ associated to a spin or spinc structure of Riemannian ...
2
votes
0answers
138 views

What are Euler density and Weyl invariants?

I would like to know as to what is the definition and significance of what are called "Euler density" and "Weyl invariants" (of weight $-d$ on a $d-$manifold) Do many (which?) of them vanish when ...
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vote
0answers
114 views

Expressing the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary

I want to express the Ricci flow as a gradient flow in a case that manifold $(M,g)$ is a Riemannian manifold with boundary. For this I use the Einstein-Hilbert action $$S(g_{\mu ...
0
votes
1answer
146 views

discrete subgroups of the isometries of a product

Suppose $X_1$ and $X_2$ are two nice metric spaces, e.g. two Riemannian manifolds, and let $G_i=Isom(X_i)$. Then $G_1\times G_2\subset Isom(X_1\times X_2)$. Suppose $X_1\times X_2$ is not compact and ...
3
votes
0answers
89 views

The Tangent Bundle of the Space of CR Structures on S^(2n+1)

Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...
4
votes
1answer
291 views

Interpetation of torsion and curvature in terms of families of nearby geodesics

Let $M$ be a Riemannian manifold with affine connection such that the metric is covariantly constant (so that the connection equals the Levi-Civita connection up to torsion). I know the ...
0
votes
1answer
162 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 ...
2
votes
0answers
118 views

Manifold with a quasi-positive curvature

As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a quasi-positive curvature to be a manifold with positive curvature. But ...
1
vote
0answers
211 views

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 ...
7
votes
2answers
272 views

Does positively curved sphere admit an isometric embedding as hypersurface in Euclidean space?

Let $(S^n, g)$ be an $n$-dimensional positively curved sphere. Assume the smoothness of the metric, does it admits an isometric embedding into $\mathbb R^{n+1}$? for $n=2$ it is proved by A.D ...
4
votes
1answer
361 views

Reverse Ricci Flow and Longtime Existence

The usual Ricci flow and normalized Ricci flow for surfaces are $$ \partial_t g = -2Kg $$ and $$ \partial_t g = -2Kg + 2sg,$$ where $K$ is the Gaussian curvature and $s$ is its average. The latter ...
6
votes
0answers
203 views

About Palais' remark that an isometry of Riemannian manifolds does not induce an isometry of the Hilbert manifolds of curves

In the paper ``Morse theory on Hilbert manifolds'' (1963), on page 326, Richard Palais makes a remark that if $\phi\colon V \to W$ is an isometry (of submanifolds of $\mathbb{R}^n$), then this does ...
10
votes
1answer
581 views

Formal adjoint of the covariant derivative

Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric. It is essentially ...
2
votes
0answers
140 views

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 ...
11
votes
1answer
220 views

Hyperbolic Manifolds which fiber over the circle

If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by ...
1
vote
1answer
158 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: ...
4
votes
1answer
332 views

Isometry group of pseudo Riemannian manifold always a Lie group? (Myers-Steenrod)

Myers-Steenrod states that the isometry group of a Riemannian manifold is a Lie group. Is that also true for pseudo Riemannian manifolds? I didn't find anything related to that. Cheers
2
votes
1answer
213 views

Regularity of metric of the double of a Riemannian manifold

Let $M$ be a Riemannian manifold with totally geodesic boundary $\partial M$. We let $\check{M}$ be its double, i.e. the disjoint union of $M$ with itself under identification of corresponding ...
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votes
1answer
219 views

Buseman function on manifolds with $Ric \ge - \left( {n - 1} \right)$

It's well known that if M is a Riemannian manifold with $Ric \ge 0$ and contains a line $\gamma $. Set ${\gamma _ + } = \gamma \left| {_{[0, + \infty )}} \right.$, ${\gamma _ - } = \gamma \left| {_{[ ...
3
votes
1answer
306 views

Shortest geodesic loop vs. shortest periodic geodesic

Are there simple conditions on a Riemannian metric on the two-sphere that imply that a geodesic loop of minimal length is actually a periodic geodesic? For example, is this true for small ...
2
votes
1answer
237 views

iwaniec's conjecture

Does anyone know whether there is any geometric applications of the iwaniec's conjecture on $ l^p $ bound of beurling alfhors transform( or the complex hilbert transform). One application could have ...
4
votes
3answers
553 views

Uniquely geodesic and CAT(0) spaces?

Improvement after J-M Schlenker's comment below : This post has been divided into two parts, the second part is here. Question : Is a finite dimensional metric space, uniquely geodesic if and only ...
6
votes
2answers
182 views

Isometric embedding as a graph

Question Let $M$ be a (finite dimensional) smooth manifold and $g,\bar{g}$ be Riemannian metrics on $M$. Under what conditions can we guarantee that there exists another finite dimensional ...
5
votes
3answers
476 views

Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?

Consider the following question: "Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?" I believe the answer to ...
0
votes
1answer
144 views

volume of a submanifold implies bounds on curvature

I would like to ask the following question: Suppose an m-dimensional manifold in an n-dimensional euclidean space, choose some point on this manifold and take an n-dimensional ball of radius R centred ...