(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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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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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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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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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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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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36 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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50 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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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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82 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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35 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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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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149 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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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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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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52 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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+200

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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63 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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44 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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36 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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69 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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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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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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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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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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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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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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51 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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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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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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68 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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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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80 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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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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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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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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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 ...
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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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53 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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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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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. ...
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Conformal Equivalence of two Riemann metrics

I'm reading a paper and encountered a concept of conformal equivalence between two Riemannian metrics on a differentiable $2$-manifold $M$ : Two Riemannian metric $g$ and $f$ are conformally ...
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Geodesics of one-dimensional manifold

I apologize if my post is "silly" because I don't know much about riemannian geometry. I know that $M = (0,1)$ (the open unit interval) can be seen as a one-dimensional manifold. Since $M$ is an ...
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2answers
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Differential Geometry Notation

When we have a metric on a manifold, there is a natural isomorphism between the tangent space and the cotangent space, and so, if I understand correctly, it is not so important to keep track of which ...
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31 views

Reference for a theorem in Riemann geometry

I am watching an online lecture by John Morgan on lektorium, and at about 33:00, he claimed a theorem which I have never heard before, that There is a formula in terms of Riemann curvature $R$ ...
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Curve on Riemannian Manifold

A curve on Riemannian Manifold is $c:I\rightarrow M$. We study many properties about it, like parallel $\bigtriangledown_\dot{c}X=0$ and geodesic $\bigtriangledown_\dot{c}\dot{c}=0$. And we apply the ...
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On a Riemannian manifold (along a geodesic), is the relation *is conjugate to* transitive?

Let $(M, g)$ be a complete Riemannian manifold. Suppose $\gamma : \mathbb{R} \rightarrow M$ is a geodesic such that the instant $0$ is conjugate to both $a$ and $b$, where the numbers $a, b, 0$ are ...
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What is the value of the 2-form $X(u(Y))-Y(u(X))-u([X,Y])$, is it $2du(X,Y)$?

Let $X,Y$ be two vector fields on a Riemannian manifold $(M,g,\nabla)$ where $\nabla $ is the symmetric metric connection on $M$ and let $u$ be a 1-form. I want to find the value or a simple form or ...
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the Definition of Connection

Let $M$ be an Riemannian Manifold and $\bigtriangledown$ be the Riemannian Connection on it. Let we think about the domain and range of $\bigtriangledown:\Gamma(M)\times\Gamma(M)\rightarrow\Gamma(M)$ ...
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Example of '$g$' which is not a metric on $S^2$.

I am trying to find out a non-degenerate, positive, bilinear form defined for every point $p$ in $S^2$, such that it is not a metric and illustrate the same (i.e. it must not be satisfying the ...
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Ruled surface defined by a Exponential map

I have a regular curve $\alpha(s)$ on a 3-D Riemannian Manifold, with Frenet frame $\{T,N,B \}$. I define a surface now as $$X_N(s,t) = exp_{\alpha(s)}(tN(\alpha(s)))$$ $exp_p$ being the ...