(Stub) The study of differential geometry with notions of infinitesimal distance given by a Riemannian metric. This allows us to consider questions about the shape of a manifold by studying its curvature.

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Riemann Sphere/Surfaces Pre-Requisites

I have recently developed a large interest in everything to do with Riemann Sphere/Surfaces. I wish to understand the topic quite well but I know that I will need to read a good number of books on ...
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49 views

Is the Laplacian $-\Delta$ on a compact manifold an isomorphism?

We know that for (a normal) domain $-\Delta:H^1_0(\Omega) \to H^{-1}(\Omega)$ is an isomorphism. What is the corresponding result for the Laplace-Bulltrami operator or more generally a Laplacian ...
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70 views

Isometry group of a compact manifold

Is an isometry group of a compact manifold always a compact group?
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1answer
56 views

Parallel translation via $e$-connection

This question is concerned with Section 2.5. of Amari and Nagaoka's Information geometry book. Let me give some background first. Let $\mathcal{P}$ be the $n$-dimensional manifold of all (strictly ...
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1answer
38 views

1-form on the Riemannian manifold

Let $\omega$ a 1-form on a riemannian manifold $(M,g)$, and for a point $x\in M$, there is a notation: $|\omega_x|_g$, what does $|\omega_x|_g$ mean?
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120 views

Show isometry of flow on a compact Riemannian manifold where the vector field is Killing

Let $(M,g)$ be a Riemannian manifold, $\nabla$ the Levi-Civita connection of $g$. A vector filed $V$ on $M$ is called a Killing field if for every $p\in M$ and every $X,Y\in T_p M$, $$ g(\nabla_X V, ...
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1answer
58 views

Find the Gauss Curvature of This Particular Metric:

Let D be an open disc centred at the origin in $ \Bbb R^2 $. Give D a Riemannian metric of the form $ (dx^2 + dy^2)/f(r)^2 $, where $ r = \sqrt{x^2 + y^2} $ and $ f(r) > 0 $. Show that the Gauss ...
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53 views

The trace of a wedge product of matrices

I'm trying understand a computation on Besse's book (p. 371). I already know the curvature operator $R:\bigwedge^2\to\bigwedge^2$ may be written in block diagonal form relative to the direct sum ...
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71 views

Is this a geodesic?

Let $(M,g)$ be a riemannian manifold. Let $p$ in $M$ and $v,v_{0}$ two vectors in $\mathrm{T}_{p}M$. I am looking at the curve $$ \gamma \, : \, t \, \longmapsto \, \mathrm{Exp}_{p}(tv+v_{0}) $$ ...
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1answer
91 views

Definition of PU(2,1)?

I know what the unitary group of complex matrices $U(n)$ is, and what $PU(n) = PSU(n) = SU(n)/(\mathbb{Z}/n)$ is. However, I found in an article mentioned $PU(2,1)$, the group of bi-holomorphisms of ...
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38 views

Notations in Riemannian Geometry

Let $f:M\rightarrow N$ be differential map. We denote tangent map $$f_*:TM\rightarrow TN$$ and cotangent map $$f^*:T^*N\rightarrow T^*M$$ Now let $M$, $N$ be Riemannian manifolds, and ...
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57 views

Metric completion of universal covering of punctured plane

It is known that the universal covering of the punctured plane $\mathbb C\setminus\{0\}$ is $\exp:\mathbb C\to\mathbb C\setminus\{0\}$. In real coordinates, $f=\exp:\tilde M=\mathbb R^2\to M=\mathbb ...
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71 views

Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
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1answer
31 views

Isometric map of geodesic

Assume a Riemann manifold $(M,g)$ and a smooth map $\sigma:M\times M\rightarrow M$, $(m_{1},m_{2})\rightarrow \sigma_{m_{1}}(m_{2})$, such that: $\forall m\in M$ $\sigma_{m}:M\rightarrow M$ is an ...
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0answers
124 views

Chain rule with covariant derivative

Let $\mathcal{M}$ be a $n$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. Consider the following function: $$\tilde{F}(v) = \operatorname{d exp}^{-1}_{p} ...
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39 views

precise meaning of connected manifold

what does it mean for a manifold to be "connected" precisely? what is the difference between a connected riemannian manifold and a nonconnected one. (i know what a riemannian manifold is a manifold ...
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75 views

connected complete totally geodesic sub manifold of $S^n$

Let $M$ and $N$ be manifolds with Riemannian metrics $g$ and $h$ respectively. A diffeomorphism $F: M\to N$ is an isometry if \begin{equation*} h_{F(x)}(T_x F(u), T_x F(v))=g_x(u,v) ...
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51 views

embedding discrete metric into manifold?

True or false: "Any edge-weighted undirected graph can be isometrically embedded into some Riemannian manifold". "isometric embedding" here means that for any pair of nodes, their shortest path ...
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91 views

Covariant derivative along curve

Let $M={\mathbb R}^{3} $ with the usual metric $g=ds^{2} =dx^{2} +dy^{2} +dz^{2} $. Let $\gamma :I\to M$ be a unit speed curve. How can I prove that $\nabla _{\gamma '} \gamma '=\gamma ''$ , where ...
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1answer
42 views

Local coordinates for two riemannian metrics

Let $(M,g)$ be a Riemannian manifold, $g' = g + f$ be another metric. Is it possible to get local coordinates such that at a point $P \in M$, $g_{ij} = \delta_{ij}$ and $f_{ij} = 0$ for all $i \not = ...
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62 views

Hamilton's Proof of the Tensor Maximum Principle

So I have more questions coming from Hamilton's "Three-Manifolds with Positive Ricci Curvature" paper. I'm working in section 9 on Preserving Positive Ricci Curvature, the proof of Theorem 9.1. I'm ...
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182 views

Taylor expansion of a vector field on manifold

In my work I have a need for some kind of analogue of Taylor expansion of a vector field on Riemannian manifold $\mathcal{M}$. I came to such an expression: $$ F(\operatorname{exp}_p(v)) = ...
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27 views

Distance under Some Metric [duplicate]

It is my homework: Let $D^*=\{(x,y)\in\mathbb R^2|0<x^2+y^2<1\}$ be the punctured unit disc in the Euclidean plane. Let $g$ be the complete Riemannian metric on $D^*$ with the constant ...
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21 views

Concerning geodesic representatives of singular homology classes on compact Riemannian manifolds

I am currently reading a book on geodesic flows where I found the following (unproved) claim: "If $M$ is a compact Riemannian manifold then any nontrivial $\alpha \in H_1(M, \mathbb{Z})$ contains a ...
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1answer
62 views

Gradient of a function restricted to a submanifold

Let $f$ be a sufficiently smooth function on a manifold $S$. Let $M$ be a sub manifold of $S$. Can someone show how is it then true that $(\text{grad}f|_M)_p$ at a point $p$ (gradient of the mapping ...
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139 views

Topology on the space of compatible almost complex structures in symplectic geometry

I have a few fairly generic questions, with a specific application to symplectic geometry in mind. Let me pose the specific problem first: Let a symplectic manifold $(M,\omega)$ be given. One is ...
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1answer
123 views

Bishop - Gromov Comparison Theorem proof and references.

I'm having trouble understanding a proof of the Bishop's volume comparison theorem and any help would be really appreciated. It's a simple part of the proof but I'm not quite getting what they want to ...
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1answer
55 views

Hilbert-Schmidt norm/smooth manifolds

Given two riemannian manifolds $M$ and $N$ and a smooth map $f$ : $M$ $\rightarrow$ $N$, we define the energy density of $f$ as the smooth function $e(f)$ : $M$ $\rightarrow$ $\mathbb{R}$ given by ...
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1answer
50 views

On Vanishing Riemann curvature tensor

If a manifold $\mathcal{M}$ has a vanishing Riemann curvature tensor, then what i) does this imply for the manifold? and ii) What is such a manifold called?
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72 views

How does curvature do that?

In his book "Riemannian geometry" Do Carmo said The curvature measures the amount that a riemannian manifold deviates from being euclidean My question is How does the curvature measure this ...
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31 views

What are the umbilic hypersurfaces in a sphere?

It is a well-known result that all umbilic hypersurfaces (complete and connected, say) of $\mathbb{R}^n$ are spheres or planes. But what can we say about umbilic hypersurfaces of a constant curvature ...
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50 views

On the definition/notation for pseudoholomorphic curves

A pseudoholomorphic curve is a map $u:(\Sigma,j) \to (M,J)$ from a Riemann surface $\Sigma$ with an almost complex structure $j$ to a manifold $M$ with an almost complex structure $J.$ We require ...
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50 views

Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...
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0answers
56 views

Inner product in projective plane

We define the projective plane as $P^2=\{[p]:\{p,-p\}\in S^2\}$ or as the set of all lines passing throught the origin in $R^3$. We define coordinates charts as page 10 in ...
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62 views

About some properties of the heat kernel

Maybe my questions are stupit, but I'm so unsure about some properties of the heat equation. Suppose we have a compact and smooth manifold M (possibly with boundary), then in a lot of books they ...
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2answers
68 views

Riemannian metric, compute

I have a question that may look for you as silly. A few years ago I took a course of Riemannian geometry. Well, the first problem I found is to understand the generalization of tangent plane (in ...
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2answers
63 views

Worst case examples of non-differentiability of the Riemannian distance function

Let $g$ be a $C^\infty$ Riemannian metric on the plane, and let $p$ be a point on the plane. Let $X$ be the set of points $x$ at which the Riemannian distance $d(p,x)$ is not differentiable. How bad ...
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1answer
45 views

affine combination of convex functions.

In a complete smooth simply connected Riemannian manifold of non-positive curvature, the square of the distance function $d^2(p,x)$ is a smooth strictly convex function of $x$. It follows that this is ...
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1answer
38 views

A connection is the limit of the newton quotient of the parallel transport

Let $E\rightarrow M$ be a vector bundle with connection $\nabla$. Denote by $\Pi_{\gamma(t_{0})}^{\gamma(t_{1})}:E_{\gamma(t_{0})}\rightarrow E_{\gamma(t_{1})}$ the parallel transport map along the ...
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99 views

Definition of the Energy of a curve

The energy of a curve $c: I \to S$ assuming S is a regular surface with a Riemannian metric $g$ is defined as : $$ E[c] = \frac{1}{2} \int_I g_{c(t)}(\dot c(t),\dot c(t))\mathsf{dt} $$ This is quite ...
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91 views

Proof of a theorem in Riemannian Geometry

Prove the following theorem: For $3\leq r\leq \infty$ let $(M; g)$ be a Riemannian $C^r$-manifold. Then there exists an isometric $C^r$-embedding of $(M; g)$ into a Euclidean space $\mathbb{R}^n$. ...
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1answer
104 views

Lie derivative on a riemannian manifold

Suppose we have a Riemannian manifold $(M,g,\nabla)$ with Levi-civita connection $\nabla$. We define a new symmetric non-metric connection $\bar\nabla$ on $M$. Then the Lie derivative of functions and ...
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1answer
30 views

Any books on isospectral manifolds?

I was searching stuff related to M.Kac's famous question "Can one hear the shape of the drum ?" I further found results due to Gordon, Webb and Wolpert in the 2D case using Sunada method. Are there ...
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1answer
155 views

What does it mean “being geodesic” is not invariant?

We know that "being geodesic" is not invariant under re-parametrization. Only affine re-parametrization preserves the property of being a geodesic. Also, a geodesic is locally distance minimizer. ...
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1answer
137 views

A new symmetric non-metric connection that generalizes the geodesic equation(Version 2)

A curve $\alpha$ on a riemannian manifold $(M,g,\nabla)$ is a geodesic if $\nabla_TT=0$, where $T$ is the tangent vector field. A generalization of this geodesic equation suggests that $\nabla_TT=\rho ...
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82 views

No constant curvature metric on $S^2 \times S^1$

I was reading the introduction to Hamilton's paper "Three-manifolds with Positive Ricci Curvature." He states that $S^2 \times S^1$ admits no metric of constant sectional curvature, and therefore ...
4
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1answer
88 views

Relation between two Riemannain connections

Let $g$ be a Riemannian metric on $M$ and let $\tilde{g}=f^{2}g$ where $f$ is a smooth function that is never zero. let $\nabla$ and $\nabla'$ be the Riemannain connections of $g$ and $\tilde{g}$ on ...
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1answer
61 views

What is Rotational on a Riemannian Manifold?

I have learned divergence, gradient and rotational in vector analysis of $\mathbb R^3$. However, when I read Riemannian Geometry, there are only definitions about divergence and gradient. So I have an ...
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1answer
62 views

Symmetry of the Riemannian curvature tensor

The Riemannian curvature tensor, in local coordinates, $R_{ijkl}$, has the following symmetries: $$R_{ijkl}+R_{jikl}=0;$$ $$R_{ijkl}+R_{ijlk}=0;$$ $$R_{ijkl}=R_{klij};$$ ...
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401 views

Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. ...