# 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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### 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

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 a certain Riemannian manifold

Notation: We denote by $\text{inj}(p)$ the injectivity radius at a point $p$ of a Riemannian manifold. By unbounded, I mean that there exist points on the manifold with arbitrarily large Riemannian ...

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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, which I know very little about.
In an attempt to prove this, I thought maybe I could show that if $M$ is compact, ...

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### The round sphere and the curvature bounded above by 1

Let $(M,g)$ be a closed connected smooth Riemannian $n$-manifold $(n\geq 3)$ and $(S^n, g_{st})$ be the standard round $n$-sphere. Suppose there exists non-zero degree map $f$, $f: (M,g)\to (S^n, g_{...

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### Sobolev embedding for $p$-forms - Frank Warner's approach

I'm trying to understand the approach of the last chapter of Frank Warner's book "Foundations of Differentiable Manifolds and Lie Group" that deals with elliptic operators on vector bundles ...

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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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### On the short distance behavior of the Green functions of powers of the Laplacian

Let $M$ be a closed Riemannian manifold and let $\Delta=dd^{*}$ be the (positive) Laplacian on $M$. Given $\lambda>0$ and a positive integer $s$, set
$G_{\lambda,s}=(\Delta^s+\lambda)^{-1}$. ...

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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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### Kazdan-Warner Type problem

let $M$ be a Riemannian manifold of dimension $n$. I am interested in the following equation:
\begin{align*}
\Delta(u)+f(x)e^{-u}=c
\end{align*}
where $f(x)\geq 0\forall x\in X$ $,f(x)$ is not ...

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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, in the flat metric ...

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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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### 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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### Existence of developable ribbonization of a surface

Let $S$ be a smooth compact surface embedded in $\mathbb{R}^{3}$. It is well-known that there exists a triangulation of $S$. I am considering an alternative way of approximating $S$, where instead of ...

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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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### 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 ...

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### Differential equation on a Riemannian manifold

Let $(M,\mu)$ be a compact Riemannian manifold of dimension $n$ and $\theta$ a differential 1-form. Write $\theta=dg+\delta\phi+h$, the decomposition of $\theta$ according to the Hodge decomposition. ...

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### Order 3 Trace-free tensor On Einstein manifold

In my recent work, I need an order 3 trace-free tensor with curvature(scalar or Ricci or other are ok) on Einstein manifold. Is there any tensor except Cotten tensor and Weyl tensor? Can people/...

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### Complete Hermitian manifolds with vanishing Chern curvature

An old theorem going back to Boothby states that a compact Hermitian manifold with Chern curvature vanishing identically is a compact quotient of a complex Lie group with a left invariant metric. Are ...

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### Statistical manifolds with trivial statistical structure after quotienting

A statistical manifold $(M,g,\nabla)$ is a Riemannian manifold with a torsion-free affine connection $\nabla$ such that $\nabla g$ is symmetric in all entries. Equivalently, there is a dual affine ...