Questions tagged [riemannian-geometry]

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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Density of zero modes

Let $(M,g)$ be a compact smooth Riemannian manifold with a smooth boundary. Let $\{(\lambda_k,\phi_k)\}_{k\in\mathbb N}$ be the spectral data on $(M,g)$, namely an orthonormal basis for $L^2(M)$ ...
3 votes
1 answer
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+100

Systems of (hyperbolic) 2nd order PDEs with lower order constraints

Certain surfaces in mechanics are endowed with the fundamental forms \begin{align} \text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\ \text{II} &= \alpha\left(\...
1 vote
0 answers
60 views

Expressing the union of principal orbits as a disjoint union of global slices for proper group actions

Setup: I was reading about slices and principal orbit theorems (Theorem 3.4.6) from these notes. Let the Lie group $G$ act on a complete Riemannian manifold $M$ properly but not necessarily freely. ...
1 vote
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Compactification of a Cartan-Hadamard manifold

Let $X$ be a simply connected manifold with nonpositive sectional curvature. It is standard that $X$ is uniquely geodesic, i.e., for any distinct points $p$ and $q$, there is a unique geodesic ...
2 votes
1 answer
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Gradient flows: evolution of geodesics

I’m trying to understand if, when I move the marginals of a Wasserstein geodesic along a contractive flow, the geodesic between the new probability measures is “near” to the geodesic connecting the ...
4 votes
1 answer
197 views

Injective hulls of metric spaces

In the context of large scale geometry and geometric group theory, I have recently come across the concept of injective hulls of metric spaces. For a metric space $X$, let $\text{In}(X)$ be the set of ...
25 votes
1 answer
732 views

Alternate proofs that hyperbolic plane can’t be isometrically immersed in $\mathbb{R}^3$

A famous theorem of Hilbert says that there is no smooth immersion of the hyperbolic plane in 3-dimensional Euclidean space. The expositions of this that I know of (in eg do Carmo’s book on curves/...
9 votes
1 answer
353 views

Perturbing metrics with nonpositive curvature

Let $M$ be a compact $3$-dimensional manifold diffeomorphic to a ball. Suppose that $M$ has nonpositive (sectional) curvature and its boundary $\partial M$ is convex, or even that $M$ is a Riemannian ...
1 vote
1 answer
106 views

Does a Riemannian submersion map horizontal geodesics to geodesics, and a relevant question?

I asked this question on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them ...
0 votes
1 answer
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Handling degenerate planes in pseudo-Riemannian geometry: impact on sectional curvature and comparison theorems

I've been studying Riemannian and pseudo-Riemannian manifolds and came across an intriguing point regarding the definition of sectional curvature in both geometries. In pseudo-Riemannian geometry, for ...
2 votes
1 answer
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Frobenius theorem and the size of integral manifold

Let $X =(X_0,X_1)\in \mathbb{R}^2$ and $Y=(Y_0,Y_1)\in \mathbb{R}^2$ be two vector fields on $\mathbb{R}^2$ such that $X,Y$ are independent on each tangent plane and $[X,Y]:=XY-YX=0$. Then by ...
6 votes
1 answer
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When is the cut locus a finite tree?

Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for ...
4 votes
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Regularity of exponential map for $C^{2,\alpha}$ Riemannian metrics

Let $g$ be a $C^{2,\alpha}$ Riemannian metric and $0<\alpha<1$. Would the exponential map $\mathrm{exp}_p$ be $C^{1,\alpha}$ as the point $p$ varies? Since $\mathrm{exp}_p$ is defined by the ...
2 votes
0 answers
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Regularity and rigidity of stable/unstable distribution for geodesic flow on noncompact negatively curved manifolds

For a volume-preserving $C^\infty$ Anosov flow on a three-dimensional compact Riemannian manifold, it was shown by Hurder & Katok that the Anosov foliations are always of class $C^{1, \alpha}$. ...
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Construct compact submanifold containing non-compact Nash embedded submanifold

$$ \newcommand{\R}{\mathbb{R}} \newcommand{\geu}{g_{\text{Eu}}} \newcommand{\X}{\mathcal{X}} \newcommand{\iX}{\mathring{\X}}$$ Let $\X$ be a closed bounded convex set in some Euclidean space. Its ...
2 votes
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What $1$-forms $\theta$ solve $\Delta \theta = f\theta$ for a smooth function $f$?

I have a seemingly basic question that I cannot find any literature on. Let $(M,g)$ be a smooth Riemannian manifold and let $\Delta:\Omega^1(M) \to \Omega^1(M)$ be the Laplace-De Rham operator on $1$-...
1 vote
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Metric of negative curvature on connected sum

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds of dimension $n\geq 2$. If we consider the connected sum $M=M_1\mathbin{\#}M_2$ of the two manifolds; can one get a smooth metric $g$ on $M$ ...
2 votes
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Growth/Decay of conformal Killing fields in cone metrics

Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric $$g= dr^2 + r^2 \gamma$$ on $[1,\infty) \times S^2$. Does there exist a nontrivial conformal Killing field vanishing ...
7 votes
5 answers
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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 ...
2 votes
1 answer
89 views

For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?

Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra ...
1 vote
0 answers
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Total curvature of a conjugate minimal surface

Let $s: S \to \Bbb R^3$ be an immersed minimal surface with finite total curvature and a proper annular end (possibly with other types of ends). What is exactly meant by a proper annular end? It is an ...
2 votes
1 answer
119 views

Parallel Jacobi fields in a Hadamard manifold

Let $M$ be a Hadamard manifold and let $c: \mathbb{R}\rightarrow M$ be a geodesic. A Jacobi field $Y$ along $c$ is called parallel if $Y'(t) = 0$ for every $t\in \mathbb{R}$. If we assume that $M$ is ...
2 votes
1 answer
285 views

Sectional curvature and injectivity radius of natural metric in cotangent bundles

In the following paper by Cielibak, Ginzburg and Kerman (arXiv link, Comm. Math. Helv. 2004 DOI link) they claim in page $3$ that the natural metric $\tilde g$ on $T^*M$ the sectional curvature is ...
5 votes
1 answer
230 views

Bochner Laplacian in coordinates

Sorry if this is a too basic question, but I didn't find an answer anywhere: The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
11 votes
2 answers
736 views

Tangent bundle of a tensor product bundle

This question was also asked here on math-stackexchange. Let $E\to M$ and $F\to M$ be vector bundles. The structure of their tangents $TE$ and $TF$ is well known. In particular, connectors map $K_E: ...
7 votes
2 answers
340 views

Elliptic regularity on manifolds: Is this true?

Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
2 votes
0 answers
55 views

For riemannian manifolds, how close can a mapping from atlas be to an isometry?

Let $(M, g)$ be an $n$-dimensional $C^k$ (or $C^\infty$) Riemannian manifold. On $M$ we can define metric $d_g$ as the infimum of lengths of curves that connect given two points. Fix $x \in M$ and $r&...
2 votes
0 answers
46 views

Smoothness of the Fréchet Function

Let $M$ be a compact Riemannian manifold and $d$ be the induced distance function. Suppose $\mu$ is a probability measure on $M$ with continuous density. The Fréchet function is defined as $$ F(x) = \...
1 vote
1 answer
224 views

On intersection of null geodesics

Let us consider a globally hyperbolic Lorentzian manifold $(M,g)$ with empty cut locus. Suppose that $p$ is a point in $M$ and consider $C^-(p)$ to be the past null cone in $M$ emanating from the ...
3 votes
0 answers
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On the linearized evolution equations in general relativity

The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
6 votes
3 answers
406 views

Closed geodesics on bumpy spheres

Main question: Does every bumpy Riemannian metric on a sphere have at least three short and prime closed geodesics, for some reasonable definition of short? E.g., a geodesic $\gamma$ could be called ...
25 votes
1 answer
1k views

Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood?

Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to ...
5 votes
0 answers
134 views

Riemannian structure on connected Hilbert manifolds

The infinite-dimensional separable Hilbert space $H$ has the unusual property that it is diffeomorphic to its unit sphere $S^{\infty}$. Therefore, $H$ admits the round metric as a complete and bounded ...
14 votes
2 answers
513 views

Variation of the Einstein Hilbert action in a coordinate-free way

I am currently writing an expository paper on gauge theory and gravity and throughout the gauge theory part I have been able to mostly stay away from coordinates unless I wished to provide specific ...
2 votes
1 answer
88 views

Simple curves on hyperbolic tori

In the paper "Simple curves on hyperbolic tori" by McShane and Rivin, they show that if $T$ is a hyperbolic once punctured torus, one can define a norm on the homology $H_1(T,\mathbb{Z})$ by ...
4 votes
3 answers
4k views

Green's function on sphere

Consider radial (normal) coordinates on a sphere $S^n, n \geq 2$. Let the "origin" be the north pole $(0, 0,..., 1)$ and the coordinates be denoted by $(r, \theta)$. We know that the Laplacian $\...
4 votes
1 answer
135 views

Can every surface be realized as a mean convex hypersurface in $\mathbb{R}^3$?

I'm wondering if every closed surface can be realized as a mean convex hypersurface in $\mathbb{R}^3$, i.e. the mean curvature vanishes or points inward. Categorizing by genus: for $S^2$ ($g = 0$) ...
5 votes
1 answer
417 views

Ricci flow negative curvature

We denote by $\mathbb{H}^n$ the hyperbolic plane of dimension $n$. I don't know much about the Ricci flow, so my first question is probably naïve : can one define a normalized Ricci flow such that the ...
1 vote
1 answer
117 views

Connectedness of fibers of almost Riemannian submersions

EDIT: Let $M,N$ be compact connected smooth Riemannian manifolds. Let us assume that $N$ is closed, while $M$ might have a geodesically convex boundary. Given $f\colon M\to N$ be an $\varepsilon$-...
0 votes
0 answers
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Does every Spin$(7)$-manifold has a unit-length spinor?

Say $M$ be a manifold with a Spin$(7)$-structure. $M$ is spin and hence spin$^c$. Say $S=S_+\oplus S_-$ be a spin$^c$-bundle on $M$. Does $S_+$ has a nowhere vanishing section? The result is true if ...
7 votes
1 answer
494 views

Conformal Killing fields satisfy a third order PDE

Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$. Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims ...
13 votes
3 answers
2k views

How to get to the earliest time zone?

You are in a plane at some point on Earth. You want to be at the earliest time zone possible at the end of your flight. What is the optimal path to take? Formally, fix spherical coordinates $(\theta, \...
1 vote
0 answers
61 views

Spin(7)-instanton

Let $M$ be a Spin$(7)$-manifold with a spin-bundle $S=S_+\oplus S_-$. There's an obvious connection on $S$ which comes from lifting the Levi-Civita connection. And it induces a connection on the ...
2 votes
0 answers
373 views

Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates

Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle $$TM \vert_{\...
17 votes
1 answer
2k views

Does anyone recognize this condition on a Riemannian metric on a vector space?

In the course of studying some oscillatory integral problems, the following strange condition came up. Let $V$ be a finite-dimensional real vector space. Let us say that a smooth Riemannian metric $...
7 votes
2 answers
1k views

Why these particular numerical factors in the definition of Gaussian curvature?

Wikipedia tells me that: Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane: $K = \lim_{r \rightarrow 0} (2 \pi r - \mbox{C}(r)) \...
3 votes
1 answer
193 views

"Almost geodesics" in Riemannian manifolds which cannot be loops

Let $M,N$ be smooth Riemannian manifolds of the same dimension. Let $0<\varepsilon<\frac{inj(N)}{100}$. Let $f\colon M\to N$ be a smooth map such that for any $x\in M$ and any $v\in T_xM$ one ...
1 vote
1 answer
83 views

When "$(\varepsilon,\delta)$-geodesic" cannot be a loop?

EDIT: Let $M$ be a smooth compact Riemannian manifold. Let $\varepsilon,\delta>0$. I call a smooth curve $\gamma\colon [a,b]\to M$ an $(\varepsilon,\delta)$-geodesic if for any $t_1<t_2<t_1+\...
24 votes
4 answers
3k views

Obtain Lorentzian manifolds from Riemannian ones by Wick rotation

In some cases, Wick rotation of a metric, formally consisting in substituting a coordinate with i times the coordinate itself, allows one to construct a Riemannian manifold starting from a Lorentzian ...
1 vote
1 answer
118 views

how to construct a finite energy map

In the construction of harmonic maps by Eells and Sampson, one needs to start with a map with finite energy and use the heat equation to deform it into a harmonic map. The construction of such a ...

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