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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Definition of a differentiable manifold and papers in Riemannian geometry

There are at least two ways of introducing a definition of differentiable manifolds. I read John Lee's excellent book "Introduction to smooth manifolds" before, but there is too much bundles there for ...
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78 views

Metric tensors. Have I got the correct understanding?

My course is covering metric tensors in a slap-dash way, so I want to ensure I have understood correctly how they are described. I hope you can confirm this! So, I believe that a metric tensor is a ...
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769 views

Riemann tensor in terms of the metric tensor

The Riemann tensor is a function of the metric tensor when the Levi-Civita connection is used. Most tensor notation based texts give the Riemann tensor in terms of the Christoffel symbols, which are ...
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92 views

A question linked with concept of Lie derivative

Suppose $M$ is a Riemannian 3-manifold. We introduce a function $t$ on $M$ such that the two dimensional surfaces "$t=\text{constant}$" in $M$ are nested topological 2-spheres with the innermost ...
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Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
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104 views

Expressions for exponential map and parallel transport

This came up in a paper I was perusing. The authors list three formulas which I have not been able to comprehend. Here M is supposed to be a simply connected space form. Then for the exponential map ...
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63 views

A question about the Euler characteristic

Let $G$ be a finite group acting freely on a compact and orientable Riemannian manifold of dimension 2. I want to show that $\chi(M/ G)=\frac{\chi(M)}{|G|}$, where $\chi$ is the Euler characteristic, ...
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238 views

Metric on an open subset of $\mathbb{R}^d$ and Christoffel symbol of the second kind

I need to prove the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the Christoffel symbol of second kind. $g^{ij}$ ...
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168 views

Covering a Riemannian manifold with geodesic balls without too much overlap

I'm looking for a proof of the following fact: Let $M$ be a compact Riemannian manifold. There is a natural number $h$, such that for any sufficiently small number $r>0$, there exists a cover of ...
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131 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
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85 views

The relation between conformally related metrics and conformal vector fields?

Two metrics $g_{1}$ and $g_{2}$ are conformally equivalent metrics if $g_{2}=e^{2\theta}g_{1}$ A vector field $X$ is called conformal if $L_{X}g=2\theta g$ where $L_{X}$ is the Lie derivative with ...
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31 views

What can we say about the integral curve of a vector field on the warped product manifold?

Let $Z=(X,Y)$ be a vector field defined on the warped product $M×_{f}N$ where $f$ is defined on $M$. The integral curve of $X$ on $M$ is $\alpha$ and the integral curve of $Y$ on $N$ is $\beta$. I ...
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122 views

Is every Lie group the automorphism group of a riemannian manifold?

Given a finite-dimensional Lie Group $G$, is there always a Riemannian manifold $M$, such that $G$ is the group of isometries of $M$?
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146 views

when can you estimate curvature from finite information about two geodesics?

Let $c_v, c_w$ be two geodesics starting at a point $p\in M$, where M is a nonpositively curved, complete, smooth Riemannian manifold. Say $c_v(\varepsilon) = \exp_p(\varepsilon v)$ and ...
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196 views

What is the importance of conformal vector fields on Riemannian manifolds?

A vector $X$ on a Riemannian manifold $(M,g)$ is called conformal if $L_{X}(g)=2sg$ where $L_{x}$ is the Lie derivative and $s$ is a real-valued function on $M$. If $s=0$, $X$ is called a killing ...
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60 views

Euler characteristic of 2-dimensional compact Lie Groups

I'd like to know why the Euler characteristic of $G$, a compact Lie Group of dimension 2, is zero. I'm aware of the fact that this is true not only for dimension 2. The point is that I'm not familiar ...
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1answer
177 views

How many degrees of freedom does a metric have on a psuedo-Riemannian manifold?

I know this not that well posed of a question so please bear with me. Suppose we have a $n$-dimensional psuedo-Riemannian manifold $(M,g)$. We have that there are $n^2$ functions that make up ...
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32 views

Validity of a simple equation

Suppose $(\Sigma,h_{ij})$ is a 3 dimensional Riemannian manifold and $S$ is a 2 dimensional submanifold of $\Sigma$. Is the following equation true? $$2(\nabla_i R^{ij})n_j=(\nabla_kR)n^k$$ where ...
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77 views

Why is this curve a topological manifold?

Why is $$M=\{(z_1,z_2)\in \mathbb{C}^2 \, |\,\, z_1^3-z_2^4=0 \}$$ a topological manifold? I understand for example why why $|z|=1$ is a topological manifold, since I can write every point as ...
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214 views

Why are affine connections called so?

I have been going through some books regarding Smooth Manifolds and Riemannian Manifolds. But I haven't been able to get an answer to one question. Could you explain what is the intuition behind ...
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1answer
58 views

generalized warped product

Let $(M_1,g_1)$, $(M_2,g_2)$ denote two Riemannian manifolds, let $(I,dt^2)$ be the unit interval with its standard metric. I would like to study the manifold $(M,g)$ where $M = I \times (M_1 \times ...
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191 views

Intersecting geodesics in a positive curvature manifold

Suppose $M$ is a connected, compact orientable 2-dimensional Riemannian manifold, with positive Gaussian curvature. I'd like to show that two non-self-intersecting closed geodesics must intersect each ...
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56 views

How to find formal adjoint operators for operators $\Gamma(E) \to \Gamma(T^*M \otimes E)$

Let $(M,g)$ be a Riemannian manifold and let $E \to M$ be a real vector bundle over $M$. Let $d_A = d+A$ be a covariant derivative on $E$. It is an $\mathbb R$-linear map $d_A \colon \Gamma(E) \to ...
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Reference on Grassmanian

Can anyone suggest a reference on the Grassmanian which describe the Riemannian structure of the Grassmanian $Gr(k, n)$? Specifically, I want to know about the geodesics, convex neigboorhood, geodesic ...
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123 views

Calculating the area and length of sets using a Riemannian metric on the sphere

Let $S^2\subseteq \mathbb{R}^3$ be the unit sphere. Let's define the Riemannian metric to be $d(x,y)=\angle(x,y)=\arccos(x,y)$. Calculate the area and circumference of the ball $B(x,R)=\left\{y\in ...
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79 views

Gradient of a funtion and inverse of metric

Having some knowledge in differential geometry on $\mathbb{R}^n$, I'm reading a book on Information geometry by Amari. Let $S=\{p_{\theta}\}$, $\theta=(\theta_1,\dots,\theta_n)$ be an $n$-dimensional ...
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2answers
131 views

A complete Riemannian manifold admits cutoff functions with uniformly bounded first derivatives

I'm reading a paper which uses the following fact; it appears to be standard but I am not sure where to look for a proof. Claim. Let $M$ be a complete Riemannian manifold (assumed to be second ...
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1answer
45 views

Compact hypersurface of $\mathbb{R}^{n+1}$ with positive curvature is diffeomorphic to $S^n$.

I have a compact hypersurface $M$ of $\mathbb{R}^{n+1}$ with positive curvature. I need to show that it is diffeomorphic to $S^n$. The hint is to consider the shape operator $A_{\nu_p} x$, where ...
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1answer
49 views

what is the $p^{*}TM$ in Chern's book : Lectures on Differential Geometry

In Chern's book, $PTM$ is the projectivised tangent boundle of m-dimentional manifold $M$, if $p:PTM \rightarrow M$ is the pulled back map, then he says $p^{*}TM$ is the vector boundle with the base ...
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74 views

Volume of a complete, simply connected Riemannian manifold of constant negative curvature

Given an $n$-dimensional complete simply connected Riemannian manifold of constant negative curvature $-1$, I need to show that $$\operatorname{vol} B_{r}(p) = \alpha_{n-1} \int_{0}^{r} ...
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58 views

A slice orthogonal to each orbit

Assume that a compact (connected) Lie group $G$ acts on a manifold $M$. We choose a $G$-invariant Riemannian metric on $M$ and a point $p \in M$. Then using the exponential map at $p$, we can obtain a ...
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80 views

Identites of Riemann curvature tensor on orthonormal frame

Suppose we consider a Riemannian manifold and a local orthonormal frame $\{Y_i\}$. I was wondering whether, for the Riemann curvature tensor $R$, there are identities with regards to expressions of ...
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38 views

On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
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346 views

Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
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105 views

Riemannian Manifolds with $n(n+1)/2$ dimensional symmetry group

Given a $n$-dimensional connected Riemannian manifold $(M,g)$, its symmetry group $G$ can be considered as a subbundle of orthonormal frame bundle of $M$ (which I call $F_OM$), yielding: $$\dim G\le ...
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1answer
66 views

Geodesics Examples

Can someone provide me exemples of connected Riemannian manifolds containing two points through each there are : (i) infinitely many geodesics (up to reparametrization) and (ii) no geodesics. ...
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110 views

The cut-off function on Riemannian manifold

Let $M$ be a complete Riemannian manifold. Can we find a constant $C>0$ so that for any $p\in M$ and $R>0, 2R<\text{inj}(M)$, we can find a function $\varphi \in C^\infty _c(B_p(2R))$, such ...
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72 views

Existence of geodesics and sign of the gaussian curvature

This is Problem 11 of this year's Miklos Schweitzer. (a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to ...
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124 views

Relation between exponential map and parallel transport

I'm starting to learn Riemannian geometry and have a question. Let $\mathcal{M}$ be a Riemannian manifold, $p \in \mathcal{M};\ \tau_{p}^{q}$ be a parallel transport from $p$ to $q$ and ...
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361 views

Homogeneous riemannian manifolds are complete. Trouble understanding proof.

I came across this proof while looking for hints on my homework, and I think it's only gotten me more confused. This is from Global Lorentzian Geometry. Lemma 5.4 If $(H,h)$ is a Riemannian ...
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324 views

Ricci curvature: step in proof of a paper by Hamilton

In Hamilton's paper "The Ricci Curvature Equation" (in Seminar on Nonlinear Partial Differential Equations, here), I can do all of Lemma 4.2 except for the following relation: $$ ...
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72 views

Why are geodesics preserved by the quotient with the isometry group $M/G$?

I'm trying to prove that if $\langle M,g\rangle$ is a riemannian manifold and $G = Isom(M)$ acts properly discontinuous on $M$, then a geodesic $c$ is send to another geodesic by the map $\pi: M ...
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140 views

Regularized distance function on Riemannian manifolds

Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let ...
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296 views

Tensors constructed out of metric other than the Riemann curvature tensor

Let $(M,g_{ab})$ be a Riemannian (or Pseudo-Riemannian) manifold and let us define a tensor field as 'something' that transforms in an appropriate way under coordinate transformations. (This is how ...
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Diffeomorphism invariant scalars of a Riemannian manifold

Let $(M,g_{ab})$ be a Riemannian manifold. I know of the following scalars that one can construct them out of the metric and its derivatives: Ricci scalar $R$ $R_{ab}R^{ab}$ $R_{abcd}R^{abcd}$ ...
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71 views

Is set of focal points of a submanifold on a normal geodesic discrete?

Let $M$ be a complete riemannian manifold, $L$ a smooth submanifold of $M$ and $\gamma$ a geodesic with $\gamma'(0)$ normal to $L$. A focal point of $L$ is a critical value of the normal exponential ...
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70 views

The control of geodesic rays

Let $M$ be a simply-connected, complete Riemannian manifold whose sectional curvature $K_M$ satisfies $-b^2\leq K_M\leq -a^2<0$. Fix two points $p,q$ in $M$, for any geodesic ray $\gamma(t)$ ...
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349 views

Negative curvature compact manifolds

I know there is a theorem about the existence of metrics with constant negative curvature in compact orientable surfaces with genus greater than 1. My intuition of the meaning of genus make me think ...
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91 views

Geodesics on pseudo-Riemannian manifolds

Consider a Riemannian manifold $M$, with a metric $g$. We can find univocally the Levi Civita connection $\nabla$ on $M$, and so a covariant derivative $D_t$ (associated to $\nabla$) along curves. A ...
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18 views

geometric intent behind defining definiteness of scalar product.

I would like to know how the definiteness/semi-definiteness(positive or negative)of a scalar product (i.e. a symmetric bilinear form) influence the geometric nature of a space on which they are ...