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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2D-metric to diagonal form with determinant 1

I wonder whether it is always possible to bring a 2D Riemannian metric to a diagonal form with determinant one by changing the coordinates, i.e. for the line element $$ ds^2 = A(x,y)\, dx^2 + B(x,...
Nikodem's user avatar
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5 votes
2 answers
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Taylor expansion of the square of the distance function on a Riemannian manifold [closed]

I have recently read the problem named "Square of the distance function on a Riemannian manifold"(enter link description here) and I am interested in the formula $ d^2(exp_{x_0}(tv),exp_{x_0}...
Luis Yanka Annalisc's user avatar
2 votes
1 answer
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If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry?

This is a cross-post. Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric. Fix $s>0$, and suppose that for any two geodesic triangles $A,B$ having area $s$, there ...
Asaf Shachar's user avatar
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7 votes
0 answers
232 views

Closed geodesics on $K(\pi,1)$ spaces

Let $M$ be a closed Riemannian manifold with non-positive sectional curvature, then it is well-known that there are no contractible closed geodesics in $M$. More generally, let $M$ be a closed ...
YHBKJ's user avatar
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Why are products of spheres integrable?

Let $n+1 \geq 3$ be an integer and $p + q = n$. Inside the unit sphere $\mathbf{S}^{n+1}$ the product \begin{equation} \mathbf{S}^p(\sqrt{p/n}) \times \mathbf{S}^q(\sqrt{q/n}) = \{ (X,Y) \in \mathbf{R}...
Leo Moos's user avatar
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4 votes
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What is the minimal length of a “Diagonal” in a Torus?

Given a Riemannian torus $(T,d)$ with fundamental group $\pi_1(T)=\langle a,b \mid ab=ba \rangle$. Denote for any $\gamma \in \pi_1(T)$ the infimum length of all representatives of $\gamma$ by $L(\...
Sebastian's user avatar
5 votes
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Can a manifold be triangulated with minimal surfaces

It is a fact stated as an exercise in chapter 9 of Lee's book "Riemannian Geometry" that any compact 2D manifold can be triangulated by geodesic triangles. Can one triangulate any compact ...
Amr's user avatar
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References for local distance approximation over Riemannian manifolds [duplicate]

Over a complete Riemannian manifold $(M,g)$, in a neighborhood of $p \in M$, the local distance can be approximated as follows: $\forall v,u \text{ unit vectors in } T_pM, \text{ and small } s, t$ $$ ...
T. W.'s user avatar
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Conditions under which a metric on a Riemannian manifold is induced by a Riemannian metric

Let $(M, g)$ be a Riemannian manifold. Lately, I've grown interested in what you may call a "modified geodesic" problem. Given some smooth, non-negative scalar field $V$ on $M$ (aptly called ...
infinitylord's user avatar
5 votes
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Examples of transitive geodesic flows that are not ergodic

What would be an easy example of a transitive geodesic flow (defined as: there is a geodesic whose velocity vectors are dense on the unit tangent bundle) that is not ergodic? Motivation. A well-known ...
alvarezpaiva's user avatar
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3 votes
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Dependence of Roe algebra and coarse index on the Riemannian metric

Let $(M,g)$ be a spin Riemannian manifold. The coarse index of the Dirac operator $D$ lies in the $K$-theory of the Roe algebra, which I will denote by $C^*(M,g)$ since its construction uses $g$. I ...
geometricK's user avatar
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6 votes
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What does it mean for the torsion to blow up?

Consider the following theorem which is the main result of the Hermitian Curvature Flow paper by Jeffrey Streets and Gang Tian: Theorem 1.1. Let $(M^{2n}, g_0, J)$ be a complex manifold with Hermitian ...
GradStudent's user avatar
1 vote
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Question in the proof of Hilbert's theorem

I'm researching about Hilbert's theorem which says that there isn't isometric immersion of a complete surface with constant negative Gaussian curvature in $\mathbb{R}^3$. I'm taking as a reference the ...
Zaragosa's user avatar
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What are we to deduce from a structure theorem of this type concerning totally geodesic maps?

I apologise in advance for the vague nature of the question, but some insight would be greatly appreciated. I'm reading a paper of Lei Ni concerning structure theorems for Kähler manifolds. Here is an ...
GradStudent's user avatar
2 votes
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118 views

Characterization of planar domains onto which a unit disk can be mapped with constant singular values

It can be shown that there are (smoothly bounded, Jordan) domains $E\subset \mathbb{R}^2$ which are $\textit{not}$ images of mappings $f$ from the unit disk (or any other planar domain), such that $\...
Daniel Castro's user avatar
3 votes
1 answer
177 views

If every non null set of geodesics intersects itself in uniformly bounded finite time, is the manifold compact?

Let $M$ be a complete, connected Riemannian manifold without boundary. Given a point $p\in M$ and a subset $K$ of $S_p M$, the unit sphere in $T_p M$, define the $K$-cone of directions $C(K)$ around ...
Nate River's user avatar
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1 vote
0 answers
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Fitting point on a Quadric curve [closed]

I am working on research project. I am currently using CloudCompare for my project, which calculates the Gaussian curvature and mean curvature to extract the geometric features of the points. I have a ...
Visal Prabhakaran's user avatar
1 vote
0 answers
38 views

Extending the Dirac operator on an open subset of a manifold and preserving positivity

Let $M$ be a spin manifold and $U\subseteq M$ an open ball. Let $D$ be the Dirac operator on $M$ with respect to some Riemannian metric $g$, acting on sections of the spinor bundle $S\to M$. Suppose ...
geometricK's user avatar
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7 votes
1 answer
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Existence of harmonic maps onto the $n$-sphere

Let $(M^n,g)$ be a closed smooth Riemannian $n$-manifold with positive scalar curvature (or positive Ricci curvature) and $(S^n, g_{st})$ be the standard round $n$-sphere. Whether there exists a non-...
Jialong Deng's user avatar
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1 vote
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Normal geodesic coordinates on submanifold comparison of coordinates

I would like to a find a formula which relates the normal geodesic coordinates associated to a submanifold to the geodesic coordinates on the manifold. More precisely, let $X$ be a closed submanifold ...
user197284's user avatar
0 votes
0 answers
113 views

A characterization of functions which Riemannian Hessian equal to zero

Consider Euclidean space $\mathbb{R}^n$, and measure distances in this space with some Riemannian metric $M(x)$. That is, for two points $x, y$, define $d(x, y)$ to be equal to $$d(x, y) = \inf_{\...
The Wind-Up Bird's user avatar
5 votes
1 answer
530 views

A corollary of the non-existence of positive scalar curvature

I've been done some work with scalar curvature and managed to give a simple proof for the following result: Let $M$ be a closed manifold which do not admit a metric of positive scalar curvature. Then ...
L.F. Cavenaghi's user avatar
2 votes
1 answer
260 views

Completeness on the tangent bundle

I was wondering if geodesics are defined for all time on compact Finsler manifolds, or more generally, for any spray on a compact manifold (where by geodesics, I simply mean the integral curves of the ...
Nikhil Sahoo's user avatar
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2 votes
1 answer
123 views

Positive scalar curvature on the double of a manifold (assuming mean convexity of the boundary)

This question is related to a previous one. Let $(M^n,g)$ be a compact Riemannian manifold with boundary. Assume it has positive scalar curvature and $\partial M$ is mean convex (positive mean ...
Eduardo Longa's user avatar
4 votes
0 answers
135 views

Eigenvalues of Laplacian and eigenvalues of curvature operator

Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
GradStudent's user avatar
4 votes
2 answers
295 views

Positive scalar curvature on the double of a manifold

Let $(M,g)$ be a compact Riemannian manifold with boundary and assume it has positive scalar curvature. Question. Is it true that $DM$, the double of $M$, admits a metric of positive scalar curvature?...
Eduardo Longa's user avatar
1 vote
1 answer
130 views

Unit Killing vector fields on pseudo Riemannian manifolds

In arXiv:math/0605371, Theorem 4 on p.8, there is the following statement: Let $X$ be a unit Killing vector field on a $n$-dimensional Riemannian manifold $M$. Then the Ricci curvature $\operatorname{...
Sand's user avatar
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2 answers
528 views

Exponential convergence of Ricci flow

I've been trying to understand the asymptotic behavior of Ricci flow, and there are two facts which I am unable to square away. I'm interested in higher dimensional manifolds, but my question is ...
Gabe K's user avatar
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1 vote
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Books and References on Geometry of Submanifold [closed]

In this semester I want to study Geometry of Submanifolds. I know Chen Bang Yen's book: Geometry of submanifolds, but it is too hard to read since its strange print. Can people recommend textbooks and/...
管山林's user avatar
2 votes
1 answer
237 views

Existence of divergence-free unit vector field in conformally rescaled euclidean metric

Question. Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div}...
Leo Moos's user avatar
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0 votes
1 answer
145 views

The relationship between the first eigenfuntions and the second eigenfuntions on sphere [closed]

Recently I considered the following question: If we give a second eigenfuntions $g$ on sphere, then can we construct a first eigenfuntions $f$ by $g$? Is there any relationship between the first ...
管山林's user avatar
2 votes
1 answer
84 views

References Request: A paper Tanno's equation

I need a paper which wrote by Tanno where he prove that a equations $f_{ijk} + k(2f_{k}g_{ij} + f_{i}g_{jk} + f_{j}g_{ik})$ can be solved if and only if on sphere. But I can not find it on Internet ...
管山林's user avatar
0 votes
1 answer
107 views

Intersection Grassmanian planes

I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...
Adam's user avatar
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0 votes
1 answer
105 views

largest geodesic ball inside a small portion of Euclidean submanifold

Suppose that $M\subseteq\mathbb R^D$ is a compact smooth Riemannian submanifold of dimension $d$, having normal injectivity radius $\tau$. Let $x_0\in M$ be a point, and $\delta\in (0,\tau)$ ...
user avatar
3 votes
1 answer
310 views

Closed manifolds of nonnegative curvature operator are symmetric spaces

In an online webinar, I heard (not directly) the statement that (closed) manifolds of nonnegative curvature operator $\mathcal{R}\geq 0$ are symmetric spaces. Is this a valid theorem? Any reference ...
C.F.G's user avatar
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1 vote
0 answers
171 views

Does there exist an isometry between a regular polygon and a circle?

In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying $$\max\{\|x-p\...
Talmsmen's user avatar
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4 votes
1 answer
280 views

Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)

It seems that there is no digital copy of Leon Karp's Ph.D. thesis L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976. on internet and his paper excerpted from his thesis is very brief ...
C.F.G's user avatar
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0 votes
0 answers
93 views

Distance Metric on a Polytope

Primary Question: Is it possible to define a distance metric on a polytope (or permutohedron in particular)? I am aware that neither is a smooth, Riemannian manifold; however, computer scientists have ...
Talmsmen's user avatar
  • 577
6 votes
0 answers
210 views

Optimal configurations on the flat torus

I'm studying (with my colleagues R.Piergallini and S.Isola) configurations of points on the flat torus which minimize an attractive or repulsive potential depending on the distance. Two model cases ...
Alessandro Della Corte's user avatar
1 vote
1 answer
162 views

Asymptotics of constant mean curvature surfaces

Let $\Sigma^n \subset \mathbf{R}^{n+1}$ be a complete, properly embedded hypersurface with constant, non-zero mean curvature $H \neq 0.$ In the case where the dimension is $n = 2$, $\Sigma$ is non-...
Leo Moos's user avatar
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1 vote
1 answer
244 views

Quotient of $\mathbb{R}^n$ by a subgroup of $\mathrm{SO}(n)$

$\DeclareMathOperator\SO{SO}$ Let $\mathcal{M}$ be an open subset of $\mathbb{R}^n$ endowed with the Euclidean metric and $\mathcal{N}$ be a Riemannian manifold. Assume that $G$ is a Lie subgroup of $\...
Chevallier's user avatar
5 votes
2 answers
373 views

On which closed Riemannian manifolds are geodesics always recurrent?

Let $M$ be a closed Riemannian manifold. What are the necessary and sufficient conditions on $M$ to ensure that for every point $p \in M$, and every geodesic $\gamma: [0, \infty) \to M$ with $\gamma(0)...
Nate River's user avatar
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1 vote
0 answers
106 views

A Kazhdan-Warner type problem

Let $X$ be a compact Riemannian manifold and I am interested in the following set of equations: \begin{align*} \Delta f+u\cdot e^{f+\lambda}=c\\ \lambda-2f=g \end{align*} where $u,g$ are given real ...
Partha's user avatar
  • 759
9 votes
1 answer
349 views

Kulkarni-Nomizu square root of the Riemann tensor

Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of ...
Carlo Mantegazza's user avatar
7 votes
1 answer
263 views

Visualizing the wave operator in two dimensions

For $n\geq 1$, let $D_n$ be the Dirac operator on the spinor bundle on the $n$-dimensional sphere $S^n$. For example, $D_1$ acts on the trivial bundle $S^1\times\mathbb{C}\to S^1$, and can be ...
geometricK's user avatar
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3 votes
0 answers
77 views

Semiconcavity estimate for the squared distance on a compact Riemannian manifold

I am currently reading this paper on the Riemannian structure of the Wasserstein space over a compact Riemannian manifold (my question doesn't concern the Wasserstein metric), specifically Section 4.1,...
grogTheFrog's user avatar
1 vote
0 answers
94 views

Computing/estimating geodesics in practice

Let's say I have a Riemannian manifold with known metric $g$. I want to compute the geodesics of this manifold, specifically with respect to the Levi-Civita connection. In practice, (i.e. with a ...
lady gaga's user avatar
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2 votes
0 answers
62 views

Bounded and Lipschitz De Rham cohomology representatives for pull-backs from classifying spaces

Let $(M,g)$ be a Riemannian manifold with $\pi_1(M)=\Gamma$. Let $\tilde{M}$ be its universal cover, and let $f\colon M\to B\Gamma$ be a classifying map. Given any smooth differential form $\omega$ on ...
geometricK's user avatar
  • 1,851
0 votes
1 answer
266 views

implicit function theorem on manifold

Suppose that $M\subseteq \mathbb R^D$ is $d$-dimensional compact submanifold with $0\in M$, having reach $\tau>0$. Thus, for every $p\in M$, $\exp_p:B_{T_pM}(p,\tau)\rightarrow B_M(p,\tau)$ is a ...
user avatar
33 votes
5 answers
6k views

How should you explain parallel transport to undergraduates?

The title is a bit deceiving, because what I really mean is the parallel transport that corresponds to the Levi–Civita connection. This is in the vein of many other questions on mathoverflow: What is ...
Andrew NC's user avatar
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