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,...
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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}...
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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 ...
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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 ...
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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}...
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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(\...
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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 ...
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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$
$$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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-...
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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 ...
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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_{\...
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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 ...
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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 ...
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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 ...
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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 (...
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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?...
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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{...
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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 ...
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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/...
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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}...
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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 ...
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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 ...
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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 ...
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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)$ ...
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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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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-...
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1
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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 $\...
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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)...
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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 ...
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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 ...
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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 ...
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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,...
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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 ...
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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 ...
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1
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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 ...
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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 ...