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Questions tagged [riemannian-geometry]

For questions about Riemann geometry, which is a branch of differential geometry dealing with Riemannian manifolds.

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Notions of Convexity

Let $(S,g)$ be a complete Riemannian manifold and $M\subset S$ an embedded submanifold (of the same dimension) with non-empty boundary $\partial M$. I am interested in understanding the relation ...
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Limit of Laplacian of the distance at the origin

Let $p$ be a point in a Riemannian manifold $M$ and $d_p$ be the distance from the point $p$. Prove that $\lim_{x\rightarrow p}\Delta d_p(x)=\infty$ I can easily prove it in $\mathbb{R}^n$. But for a ...
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$M^n\subset \mathbb{R}^{n+1}$ isometric immersion, $R\equiv 0\Rightarrow M$ locally isometric to $\mathbb{R}^n$?

Let $f:M^n\to\mathbb{R}^{n+1}$ be an isometric immersion with $R\equiv 0$, where $R$ is the curvature tensor of $M$. Can we say that $M$ is locally isometric to $\mathbb{R}^n$? I know that by ...
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Show that $[\widetilde{X},\widetilde{Y}]^H=\widetilde{[X,Y]}$ and that $[\widetilde{X},W]$ is vertical if $W$ is vertical.

This is problem 9 from chapter 5 in Riemannian Manifolds: An introduction to Curvature. Suppose $p:(\widetilde M,\widetilde g)\rightarrow (M,g)$ is a Riemannian submersion. We denote $\widetilde X$ ...
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A basic question on mutually orthogonal coordinate systems

I am reading the first chapter of Information Geometry and its applications by Amari. I am struggling to grasp a basic concept about mutually orthogonal coordinate systems. Since the book is not ...
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$\Delta_g f = 0$ on the Riemannian Manifold $(\mathbb{R}^3 \setminus B , g)$ with conditions on boundary and at infinity

Consider the manifold $M = \mathbb{R}^3 \setminus B$ where $B$ is the ball with radius $1$ with riemannian metric $g$ (not necessarily the euclidean metric). I am looking for solutions to $\Delta_g ...
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How to find an embedded submanifold in $\mathbb{R}^{n}$ which pass through some given points in $\mathbb{R}^{n}$ and has the lowest dimension?

How to find an embedded submanifold in $\mathbb{R}^{n}$ which pass through some given points in $\mathbb{R}^{n}$ and has the lowest dimension? I have no idea with this question. I know, at least, the ...
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Computing all possible conformal factors on the sphere

Proposition to prove. Let $\tau\colon \mathbb S^n\to \mathbb S^n$ be a conformal map, meaning that $\tau^\star g= \Lambda^2 g$ for a scalar field $\Lambda$. (Here $g$ denotes the standard metric ...
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Is the warped product: $dr^{2} + r^{2} \gamma$ always flat in 3 dimensions?

Suppose you have a Riemannian 2-sphere $(S^{2} , \gamma)$. Define a metric $g$ on $M = S^{2} \times [1,\infty)$ in this way: If $(x_{1}, x_2)$ is a coordinate chart on $S^{2}$ then $$g = dr^{2} + r^{...
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How to prove that the equation is not possible

I came across another very complex equation (calculating the Gaussian curvature of a surface): \begin{align*}\frac{-m}{2}=&(2A^2+A)(Du+k)^3u^{(3+6A+4B)}\\ &+AD(Du+k)^2u^{(6A+4B+4)}\\ &+(...
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Finding curvature of a surface of revolution with metric inherited from $\mathbb{R^3}$

I have a curve in $\mathbb{R^3}$ as $z= f(x)$ in the $xz$- plane and we let $S$ be the surface generated from this curve in the space by revolving it about the $z$- axis. Now, I'm asked to find it's ...
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Weitzenböck identity for $TM$-valued differential forms

Let $M$ be a Riemannian manifold, and let $\nabla$ denote its Levi-Civita connection. We have two second order differential operators $\Gamma(T^*M \otimes TM) \to \Gamma(T^*M \otimes TM)$: The ...
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Definition of $\ell$-dimensional points [Geometric Measure Theory]

Reference: Cheeger, J., Colding, T., _On the structure of spaces with Ricci curvature bounded below., J. Differential Geometry, 45 (1997) 406--480. I have a question regarding the following ...
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Condition for Riemannian distance to be equal to metric distance

If $M$ is a metric space than it is a topological space and if it is locally homeomorphic to to $R^n$ we say that it is a manifold and if we equipped this manifold with a inner product $g_p$ on ...
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Can someone help in computing curvature tensor of a surface?

I have a problem, I've been thinking about all day. Came across this while browsing some lecture notes online. So, I have a surface in space say, described as $z= f(x,y)$ and I want to find it's ...
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Writing a vector field as the gradient of a function defined on a Riemannian manifold.

I'm reading this paper and there is a step in theorem $3.1$ on pages $4$ and $5$ that I can't understand. Basically, my doubt is how to ensure that there is a potential function for the vector field $...
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Is $SL(n,\mathbb{R})/SL(n, \mathbb{Z})$ a Hausdorff space?

The special linear group $SL(n, \mathbb{R})$ of degree $n$ over $\mathbb{R}$ is the set of $n \times n$ matrices with determinant $1$, with the group operations of ordinary matrix multiplication and ...
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Geodesics on a Riemannian manifold under non-Levi-Civita connections

I'm a beginner on this topic—so please comment if anything is ambiguous, unclear, or wrong. In particular, I'm trying to figure out how to think of geodesics under arbitrary connections. Background ...
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Intuition of cocycles and there use in dynamical systems

I’ve come across several papers and lectures that use cocycles to talk about dynamics on a manifold. However, I haven’t come across an actual definition of what a cocycle is. Could someone give a ...
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Covariant derivative of determinant of the metric tensor

Let $(M,g)$ be a Riemannian manifold and $g$ the Riemannian metric in coordinates $g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$, where $x^{i}$ are local coordinates on $M$. Denote by $g^{\alpha \...
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Riemannian distance via exponential map

Let $(M,g)$ be a Riemannian manifold, compact if need be. Take an arbitrary $(p,v)\in TM$ and consider the geodesic starting at $p$ in direction $v$, i.e., $\gamma\colon I\to M, t\mapsto \exp_p(tv)$. ...
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Question about a proof on Spivak's Comprehensive Introduction to Differential Geometry

On the Addedum to the Chapter VI, vol. II, the first proposition states that to connections have the same geodesics if and only if their difference tensor is antisymmetric. When proving that if the ...
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Example $4.8$ - Chapter $0$ - Do Carmo's Riemannian Geometry

I have some doubts about the proof in example $4.8$ in chapter $0$ of Do Carmo's Riemannian Geometry book concerning to the part of the proof that the canonical projection of a differentiable manifold ...
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The pushforward of transition function of an orientable Riemannian manifold is in $SO(n)$?

Based on this notes last line, the pushforward of transition function of an orientable Riemannian manifold of dimension $2$ is in $SO(2)$. I wonder why it is true.
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Understanding a point of the example that the Euclidean metric is a Riemannian metric

A hint that many geometers give for people who start in Riemannian Geometry is associate the definitions of the course of Differential Geometry of curves and surfaces on $\mathbb{R}^3$ with the ...
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Riemmanian distance on $\mathbb{R}$

From Riemannian geometry, the distance between two points $x$ and $y$ on a Riemmanian manifold $M$ is given (abstractly) by $d(x,y)= \inf \lbrace \ell (\gamma) : \gamma \in \Omega_{x,y} \rbrace$, (1) ...
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Clifton-Pohl Torus and $\Gamma$ acting properly discontinuous

Problem: Let $M = \mathbb{R}^2 \setminus \left\{ (0,0 \right\}$ be the pseudo-Riemannian manifold with metric $$ ds^2 = \frac{2 du dv}{ u^2 + v^2}. $$ Let $\mu(u, v) = (2u, 2v)$. This is an isometry (...
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Laplacian of distance function

Suppose $\Sigma$ is a hypersurface in the complete Riemannian manifold $M$ with positive ricci curvature. Denote the distance function to $\Sigma$ in $M$ by d. Now for small distance $c$, consider the ...
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How gradient transform under homotety

Let $\Sigma^n \subseteq \mathbb{R}^{n+1}$ be a smooth hypersurface. Let $\lambda > 0$ be a constant and let define $\tilde{\Sigma} := \lambda \Sigma$. Let $f$ be a smooth function on $\Sigma$. ...
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Why are Compact Steady and Expanding Solitons always Einstein Manifolds? [duplicate]

I am reading some lecture notes on Harnack inequalities and the Ricci flow and at one point the author remarks that it is important that the definitions for gradient steady, expanding and shrinking ...
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classify the plane projective curve $C\subset \mathbb{P}^2$ defined by $X^n+Y^n-Z^n$ with $n\in\mathbb{N}$.

Classify the plane projective curve $C\subset \mathbb{P}^2$ defined by $X^n+Y^n-Z^n$ with $n\in\mathbb{N}$. I need to know how the aforementioned curve behaves when $n$ is even or odd, if it is ...
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Einstein metrics on spheres

I've got a couple of quick questions that came up after reading a peculiar statement in some article. The sentence says something like "... is the $N$-dimensional sphere with constant Ricci curvature ...
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Variation of the Christoffel Symbols

I am reading some lecture notes on Riemmanian geometry where it states if we take $\partial_s g_{ij} = h_{ij}$ to be the variation of a Riemannian metric and choose normal coordinates for $g$, the ...
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1answer
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Computation of the Laplacian form of $g_{\sigma, \eta}$ in Huisken and Sinestrari's paper

I'm trying understand how to compute evolution equation for $g_{\sigma, \eta}$ defined on page $3$ of this article and I'm reading the Lectures on Mean Curvature Flows by Xi-Ping Zhu, which gives some ...
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Can this integral be made non-positive?

Let $M \subset \mathbb{S}^3$ be a closed, connected and orientable embedded (and minimal, if important) surface. Choose a unit normal vector field $\eta: M \to \mathbb{S}^3$ along $M$ and a point $p_0 ...
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Reference for Uniformization Theorem

I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincare's Uniformization Theorem at a basic level. Any good powerpoint notes, short ...
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Dupin Cyclide: Cartesian coordinates to parametric coordinates

I have been given points in Cartesian coordinates that lie on Dupin's cyclide. I am simply trying to extract the corresponding parametric coordinates. Given two parameters $u,v \in [0,2\pi]$, the ...
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Relation between Levi-Civita connection and any another metric connection.

On a Riemannian manifold $(M,g),$ we have $D^g,$ Levi-Civita connection, the only connection that is metric (i.e. $D^g g=0$) and without torsion (i.e. $T^{D^g}=0$). My question is that if I have say $...
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Second derivative of rotation field, Derivation of bendings and torsion

I consider a smooth rotationfield $R:[0,\ell]\times (-\varepsilon, \varepsilon) \rightarrow \operatorname{SO}(3)$, $x \in [0,\ell],\, t \in (-\varepsilon, \varepsilon)$. Then the map $R^T \...
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How to show if $X$ is Killing field then it is tangent to the geodesic spheres centred at a point $p$?

Let $M$ be a Riemannianiam manifold with Levi-Civita connection and $X $ be a smooth vector field on $M$. Let $\phi : (-\epsilon, \epsilon) × V \to M$ be the local flow of $X$ in $M$. Problem is- if $...
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Explicit construction and computations on higher genus Riemann surfaces

I'm learning about the higher genus ($g>1$) Riemann surfaces and I find it hard due to the lack of explicit examples. Specifically I'm interested in the basis of holomorphic forms, Abel map, prime ...
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Writing a coordinate formula for $|df|^2$

I need to write a coordinate formula for $|df|^2$, where $f$ is a smooth function. This is in the context of Riemannian Geometry. I know that $|X|^2$ can be written as $\langle X, X\rangle$, which ...
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1answer
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Condition on the existence of a spherical triangle

It is known that $a,b,c>0$ are the sides of a triangle in the Euclidean plane if and only if $$a+b>c,\hspace{0.3cm} a+c>b,\hspace{0.3cm} b+c>a.$$ I would like to give a similar condition ...
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Computation the shape operator of a graph of function

Note. My question is based on Lee's Introduction to Riemannian Manifolds, exercise 8-1. Let $f: \mathbb{R}^n \to \mathbb{R}$ be a smooth function. Its graph $$ S := \{ (x, f(x)) : x \in \mathbb{R}^...
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How to see the Lie derivative of the tensor metric $g$ in terms of the Levi-Civita connection

According the first line on page $2$ of this paper, A smooth vector field $\xi$ on a Riemannian manifold $(M, g)$ is said to be a conformal vector field if its flow consists of conformal ...
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Stokes theorem and Volume forms

I have the following short argument which seems to say that there are no non-vanishing, (n-1)-forms on a closed manifold of dimension n but I am not very confident in my understanding of a "volume ...
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Lemma $3.2$ - Mean curvature flow singularities for mean convex surfaces

This is a lemma of the paper "Mean curvature flow singularities for mean convex surfaces" by Gerhard Huisken and Carlo Sinestrari (the paper is available here): $\textbf{Lemma 3.2.}$ Suppose $(1 + \...
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Integral of a particular vector field on manifold

This question is a better reformulation of a part of this question. Suppose $M$ is a Riemannian manifold and let $y \in M$ be a fixed point. Let $$g: M \to T_yM$$be a smooth function. Does it make ...
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Can someone explain the motivation behind 'normal neighborhood, normal coordinates' of a point in Riemannian manifold?

I am studying on my own from do Carmo's Riemannian-geometry text and in chapter 3, the author introduces 'normal neighborhood of a point in Riemannian manifold $M$ and normal neighborhood of $0 \in T_{...