A branch of differential geometry dealing with Riemannian manifolds. *Riemannian manifolds* are smooth manifolds with an inner product smoothly attached to the tangent space of each point. Usually, Riemannian geometry focuses on the notions of distance, curvature, and shape. Consider using this tag ...

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Invariance of function under isometry of Riemannian manifolds

Suppose that $(M,g)$ and $(N,g')$ are Riemannian manifolds and that $f: M \to N$ is an isometry. Now take smooth vector fields $X, Y, Z$ on $M$. Is it true that $X\langle Y, Z\rangle_p = ...
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1answer
21 views

On conformal metrics notation

A simple question, just for clarifying: suppose we have two riemannian metrics $g$ and $\tilde{g}$ in a differentiable manifold $M$, and assume they are conformal say, with $\tilde{g} = \mu g$ for ...
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1answer
26 views

Estimate of the Laplacian of a distance function on a Riemannian manifold with non-negative Ricci curvature

I'm reading the Petersen's book "Riemannian Geometry". My goal is to learn the proof of the Cheeger-Gromoll Splitting Theorem. It's quite complex and made by a lot of step. Now I'm trying to ...
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2answers
56 views

Is Gaussian curvature intrinsic in higher dimensions?

Let $M$ be a hypersurface (a submanifold of codimension 1) in $\mathbb{R}^{n}$. Is it true that it's Gaussian curvature intrinsic? (when $n>3$). Reminder: We focus our attention on a small part ...
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1answer
30 views

Connect of exchange vector field .

$D$ is Levi-Civita connect. $U,V,W,Z$ are vector field. $S$ is (0,4) tensor. And $D^2_{X,Y}S=D_XD_YS-D_{D_XY}S$. Are there any easy way to compute the below equation ? If I unfold it, it is ...
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36 views

Riemannian surface, identity relating scalar curvatures and Laplacian [closed]

Let $S$ be a Riemannian surface, i.e. a $2$-dimensional manifold, with metric $g$. Define a new metric $\tilde{g}$ by $\tilde{g} = e^fg$ for some smooth function $f$. If $s_{\tilde{g}}$ and $s_g$ are ...
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1answer
67 views

Is a Kähler manifold necessarily symplectic?

Let $M$ be a Riemannian manifold. If we pick a basepoint $p \in M$, then for any smooth path $\gamma: [0, 1] \to M$, parallel transport along $\gamma$ induces an automorphism $g_\gamma \in ...
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Global Chart implies no cut locus?

If a manifold $M$ admits a global chart, does this imply that there exists a point $p\in M$ such that $Cut_p=\emptyset$? Recall: Definition of $Cut_p$: Let $\mathfrak{C}_p$ be defined as the set ...
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Single atlas implies global normal coordinates?

Let $M$ be a class $C^k$-Riemannian manifold and suppose there exists an atlas $\langle U,\psi\rangle$ for $M$ containing only one global chart. Does this imply that the Riemmanian ...
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1answer
59 views

Riemannian manifold, $\alpha \in \Omega^p(M)$ parallel implies $\alpha$ is closed?

Let $M$ be a Riemannian manifold, and let $\alpha \in \Omega^p(M)$ be parallel; i.e. suppose $\nabla \alpha = 0$ where $\nabla$ is the Levi-Civita connection. Does it necessarily follow that $\alpha$ ...
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32 views

Injectivity radius of Exponential and curvature

Define : Injectivity radius , Exponential This question is considered in Riemann manifold. I think the Injectivity radius is connect with curvature. I guess the Injectivity radius can be ...
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1answer
34 views

Flat Extrinsic Vs. Intrinsic Distance

Context: Let $\Psi: \mathbb{R}^d \rightarrow \mathscr{H}$ be a $C^k$-embedding of $\mathbb{R}^d$ into a Hilbert space $\mathscr{H}$. We may view $\mathscr{M}:=Im(\Psi)$ as a submanifold of ...
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19 views

Relation between homomorphisms and Lipschitz functions

Given a Riemannian (say, connected) manifold $(M,g)$ one can produce the metric via a formula $d(x,y)=\inf_{\gamma}l(\gamma)$ where $\gamma$ is piecewise smooth curve joining $x$ and $y$. There is ...
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1answer
39 views

Implicit formula for the Levi-Civita connection

Let $(M, g)$ be a Riemannian manifold and $X, Y, Z$ smooth vector fields on $M$. Let $\theta_X$ be the $1$-form defined as $\theta_X(Y) = g(X,Y)$ and let $d\theta_X$ be its exterior derivative. Let ...
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0answers
20 views

Inequality among Hessians

I know that this could be a stupid question but I would like to be sure on this point. I need to study the Maximum Principle on Petersen's Riemannian Geometry. This is the first Lemma. Let $ ...
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0answers
92 views

The cohomology of the Dirac operator $d+d^{*}$

Let $(M,g))$ be a Riemannian manifold with the Hodge dual operator $d^{*}$. Is there a name (and some computation in some reference) for the cohomology of the complex of Harmonic forms with ...
11
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1answer
161 views

Is geodesic distance equivalent to “norm distance” in $SL_n(\mathbb{R})$?

Take any norm, $\|\cdot\|$on $\mathbb{R}^n,$ and consider the resulting norm on $SL_n(\mathbb{R})$: $$\|A\|:= sup\{\|Av\|: \|v\|=1\}.$$ Now take any left-invariant Riemannian metric, $g$, on ...
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13 views

Restricted root descomposition of $SL(n,R)/SO(n).$

I am trying to find the restricted root space descomposition asociated with the Rimannian symmetric space SL(n,R)/SO(n) but I am getting a little bit stucked. Could someone help me? Many thanks.
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24 views

Definition of the Hessian on Rieamannian manifolds

I can't understand the following definition given by Petersen in his Riemannian Geometry book. Let $M$ be a Riemannian manifold and let $f \colon M \to \mathbb{R}$ be a smooth function. Let ...
2
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1answer
46 views

Definition of tensors

I'm studying the book "Riemannian Geometry" by Petersen and since I'm new to the subject, I'm helping myself also with the more introductory DoCarmos's book. I'm a bit confused about the definition ...
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15 views

how to subtract mean from a set of SPD matrices

I have a set of SPD matrices and I know how to calculate their mean. My question is: Is there any method to subtract the mean from each sample? In Euclidean space we simply subtract the mean ...
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19 views

Why do we call it “structure equation”?

In Riemannian geometry there are so-called structure equations by E.Cartan,the question is simple:why do we call them structure equations?What kind of structure is it?
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31 views

Spectrum of Laplacian of a compact PseudoRiemannian manifold

Is the Spectrum of Laplacian of a compact PseudoRiemannian manifold discrete? I know that it should be for Riemannian manifold, but I can't find anything about Pseudo-Riemannian manifold. Does ...
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125 views

Flatness of torus and surfaces of higher genus

For the very first sight it may be surprise that the ordinary torus $S^1 \times S^1$ is flat: one argument to see this is the following. One can imagine a torus as a square with opposite sides ...
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The furthest point to this torus

Recall that the (geodesic) distance on the unit sphere $S^n$ is given by $$ d(p, q) = \arccos \langle p, q \rangle. $$ Let $f_r = f : \mathbb{R}^2 \to S^3$ be defined by $$f(\theta, \phi) = \left(r ...
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3answers
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definition of Riemannian metrics in do Carmo

The following is the definition of Riemannian Metrics in Riemannian Geometry by do Carmo: I don't quite understand the underlined sentence. In the book, the following is the definition of ...
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25 views

Why Laplace-Beltrami operator is so popular for 3D shape analysis.?

Apart from providing orthogonal basis in form of eigen functions what is the reason that Laplace-Beltrami operator is so popular in shape and point cloud processing.
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42 views

Literature Request: Stochastic Differential Geometry

I've in my studies taken (introductory, at the masters level) courses on both stochastic calculus, differential geometry (both elementary at the level of Pressley's book, and more advanced at the ...
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1answer
28 views

Compact GX-manifolds

Let $M=(G,X)$ be a compact smooth Lorentzian manifold with constant sectional curvature, where $X$ is any of the well-known spaceforms $\mathcal{M}^n$, de-Sitter och Anti-de-Sitter and G is their ...
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55 views

Where do they study Riemannian Geometry in Europe? [closed]

I would like to satisfy this curiosity. Which are the European universities that pay more attention and efforts in to research in Riemannian Geometry?
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Can I prescribe the geodesics?

Consider $J$ an open interval of $\mathbb{R}$. An inner product on $\mathbb{R}$ is necessarily of the form $(u,v) \in \mathbb{R}^{2} \, \mapsto \, auv$ with $a > 0$. Therefore, a Riemannian ...
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1answer
66 views

Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure

$\newcommand{\diff}{\mathrm{d}}$ TL;DR Having read this I know something about Haar measures, in particular that a left-invariant one exists and is unique on any Lie group $G$. I know that defining: ...
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1answer
27 views

Explicitly constructing the Total Metric on a line bundle

Suppose that I'm given a Riemmanian manifold $<B,g_B>$ and a real line bundle $\pi:E\rightarrow B$, such that each fiber above any $b\in B$ comes equipped with a Riemmanian metric $g_b$. Then ...
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1answer
34 views

Is every totally geodesic surface of $\mathbb{R}^3$ contained in a plane?

Is it true that every totally geodesic surface $S$ of $\mathbb{R}^3$ is contained in a plane? Locally, this is clear since geodesics in $S$ must be straight lines, and locally the exponential map is ...
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1answer
36 views

Incompleteness of Lorentzian manifolds

Let $\tilde{M}$ be a simply connected Lorentzian manifold and suppose that $\tilde{M}$ admits some Riemannian metric. Question: What can be said about the relation between the geodesic completeness ...
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1answer
27 views

Simultaneous diagonalization of two Riemmanian metrics

Let $M$ be a smooth manifold and $g,h$ two Riemannian metrics on $M.$ Can one find a local frame $(e_i)$ orthonormal with respect to $g$ such that $$ h(e_i,e_j)=0$$ $\forall i \neq j?$
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If the covariant derivative vanishes

suppose $(M,g)$ is a Riemannian manifold and $\nabla$ denotes the Levi-Civita Connection on $M$. If the covariant derivative $\nabla R$ of the Riemannian curvature tensor $R$ vanishes, i.e. $\nabla ...
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Eigenfunctions of the Dirichlet Laplacian in balls

I am trying to find out about the Dirichlet eigenvalues and eigenfunctions of the Laplacian on $B(0, 1) \subset \mathbb{R}^n$. As pointed out in this MSE post, one needs to use polar coordinates, ...
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1answer
33 views

Divergence of a Tensor (Proof verification)

Let $M$ be a Riemannian manifold with metric given by: $ds^2=-f(r)dt^2+h(r)dr^2+r^2(d\theta^2+sin^2\theta d\phi^2)$ Let $T^{\mu \nu}$ be the tensor defined by: $T^{\mu ...
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1answer
32 views

Ricci Tensor and contraction.

I have a very nice description of the Riemann tensor in the form $R_{\mu \nu \rho \sigma}$ only depending on the second derivatives on the metric. Given $g^{a \mu}R_{a \nu \rho \sigma}=R^{\mu}_{\nu ...
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2answers
94 views

Divergence in Riemannian Geometry (General Relativity)

I'm taking a course in General Relativity and I'm having some problems with the notation. I know that Einstein's tensor verifies $\nabla_aG^{ab}=0$. In physics textbooks this consequence of Bianchi ...
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How do conformal maps affect curvature?

Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemannian connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with ...
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1answer
36 views

Local isometry between simply connected manifolds

Suppose $D:\tilde{M}\rightarrow N$ is a local diffeomorphism between two simply connected smooth manifolds $\tilde{M}$ and $\tilde{N}$. $D$ is onto. In the case of $D$ being a covering map, it follows ...
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1answer
35 views

Intuitive understanding into the mean curvature flow

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $$S = ...
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2answers
30 views

The configuration space of directed great circles on the sphere

I am reading a proof of Pu's inequality in Katz' book "Systolic geometry and topology". In section 6.4, he describes a fibration $q : SO(3) \to S^2$ which I can't quite figure out. First, he thinks ...
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35 views

How to define and compute the norm of a vector with riemannian metric?

Let us consider for example, the riemannian metric $g=e^xdx^2+dy^2$ (it is symmetric and definite positive), with associated matrix $\begin{pmatrix} e^x & 0\\ 0 & 1 \end{pmatrix}$. Consider ...
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1answer
57 views

The Levi-Civita Connection on the Hyperbolic Plane

In this question here, I asked about computing the Levi-Civita connection matrix on the Hyperbolic Plane, defined as $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = ...
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1answer
44 views

Linear Connection on the Hyperbolic Plane

For the upper half-plane $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = \frac{1}{y^2}(dx^2+dy^2)$, I computed the Christoffel symbols as follows: ...
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1answer
25 views

Schur's theorem in DoCarmo's “Riemannian Geometry”

The exercise 8 of chapter 4 of Do Carmo's "Riemannian Geometry" ask to prove the Schur's Theorem. I don't understand a step in the hint (the "hint" is essentially the proof of the theorem). Schur's ...
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1answer
41 views

Sectional Curvature, Gauss curvature

I have a problem with a computation which shows that the sectional curvature coincide with the Gauss Curvature in dimension 2. This is the definition of sectional curvature I am using: ...