(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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6
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1answer
63 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
3
votes
1answer
41 views

Klein Bottle Embedding on $\mathbb{R}^4$.

First of all, I am aware of the question in How to embed Klein Bottle into $R^4$ , which was inconclusive. Anyway, I've made some progress, but I still have a question. I am using Do Carmo's ...
1
vote
1answer
12 views

The higher-order estimates for the distance function

Let $M$ be a complete Riemannian manifold such that inj$(M)\geq l>0$ and $|\nabla^k\text{Rm}|\leq A_k$ for any $k \geq 0$ . For a point $p$ on $M$, we have a distance function $r(x)=d(x,p)$. For ...
1
vote
0answers
13 views

Scalar Curvature of a metric on the hemisphere, from a paper on the Min-Oo Conjecture

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition: Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...
4
votes
1answer
37 views

Schwarzschild metric tensor normal vectors

The Euclidean Schwarzschild metric describing a manifold (a black hole, though this is not relevant to the question) is given by, $$\mathrm{d}s^2 = \left( 1-\frac{2GM}{r}\right)\mathrm{d}\tau^2 + ...
3
votes
1answer
49 views

The Lie derivative by an infinitesimal action is invertible at an isolated zero point

Let a compact Lie group $G$ act on a manifold $M$. Fix $X \in \mathfrak g$ and we write $X_M$ for the infinitesimal action of $X$. Assume that $p \in M$ is a zero point of $X_M$. Define a linear map ...
3
votes
1answer
25 views

Chain rule quesition: proving that the Weingarten map is self-adjoint

I'm reading through the proof in this paper (http://www.math.leidenuniv.nl/scripties/JaibiBach.pdf) but I'm stuck at the line: "Using the chain rule we get: $L_p(\phi_v) = -Dn(\phi_v) = - \frac ...
3
votes
1answer
64 views

Tensor Laplacian

For a general tensor $T_{\mu_1 \dots \mu_n}$ on a (pseudo-)Riemannian manifold, is it true that $\Delta (T_{\mu_1 \dots \mu_n})= (\Delta T)_{\mu_1 \dots \mu_n}$? In general, it is not true that ...
5
votes
1answer
66 views

Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
1
vote
1answer
56 views

A lemma is John Lee's Riemannian Manifold having problem with proving it

The tangential connection on an embedded submanifold $M ⊂ R^n$ is symmetric. the hint is let $X,Y$ be vector fields that are tangent of M at points of M, so is $[X,Y]$ I start with $$T(X,Y)=\nabla_x ...
0
votes
0answers
16 views

Variation on Fubini's Theorem

My attempt: Let $P_1$ be a regular partition of $R_1$ and $P_2$ a regular partition of $R_2$. Denote by $P$ the corresponding regular partition of $R_1\times R_2$. Given a generalized rectangle ...
1
vote
0answers
17 views

Definition of a Regular Partition of a Closed Generalized Rectangle in $\mathbb R^n$

What the heck does this definition of a regular partition $P$ of $R$ mean? I follow what it is saying until we get to the last part, "the $k_1\cdot k_2\cdot \cdots \cdot k_n$ subrectangles of the ...
1
vote
0answers
24 views

Volume form for $SE(n)$ and/or $E(n)$. [duplicate]

I wonder what happens when you construct the Tiling spaces considering the natural action of $SE(n)$ or $E(n)$ rather than $\mathbb R^n$. In order to do that, I need to understand both the riemannian ...
0
votes
0answers
20 views

is there any relationship between the convexity radii of two “near” points in a riemannian manifold?

For example, if the convexity radius of a point x in a riemannian manifold M (without boundary) is R, what can we say about the convexity radius of points in B_R(x)? The convexity radius of x is the ...
1
vote
0answers
38 views

Is $\nabla_{X}:\Gamma(E) \to \Gamma(E)$ continuos aplication?

Be $\Gamma(E)$ smooth sections space of an vector bundle, $\Gamma(E)$ with Fréchet topology. Gived a conexion in E and a smooth vector field $X$, Is $\nabla_{X}: \Gamma(E) \to \Gamma(E)$ continuos ...
0
votes
1answer
33 views

Orthogonal connection on tangent bundle

What does orthogonality of connection mean in coordinate way? As I understand, a connection $\nabla: \Lambda^1M \rightarrow \Lambda^1M \otimes \Lambda^1M$ is torsion-free iff in any local coordinates ...
1
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0answers
57 views

Calculus on manifolds

I am studying manifold theory ( smooth manifold). As I study the subject, I feel more and more confused. I am not yet at chapter dealing with deferential forms and integration. I can understand that ...
2
votes
2answers
54 views

Nonlinear PDE from Riemannian Geometry

I am wondering if anyone knows an approach to finding solutions to the following PDE: $-e^{-2u}\Delta u=\alpha$. Here $u=u(x,y)$ is an unknown real-valued function of 2 variables and $\alpha$ is a ...
1
vote
0answers
46 views

Level Sets and Derivative

Suppose that you are given two functions, $u$ and $v$ of two variables, with $u(0,0)=v(0,0)=0$. You know that for a large enough $n$, the $n$-th Differentials are different at the origin: if $n=1$, ...
1
vote
0answers
148 views

Geodesic question

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability distributions on $\mathcal{X}=\{x_0,\dots,x_n\}$. Each $p=(p(x_0),\dots,p(x_n))\in ...
0
votes
0answers
25 views

Connection on $\operatorname{Spin}^\mathbb{C}$ spinor bundle

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} \oplus ...
1
vote
2answers
55 views

Forms on Riemann Surfaces

I want to show that the space of smooth $(1,0)$ forms on a compact Riemann surface $X$ has the natural splitting: $\mathcal{E}^{1,0}(X)=\Omega(X) \oplus \partial\mathcal{E}^{0}(X)$, where $\Omega(X)$ ...
0
votes
0answers
30 views

Question concerning $e$-geodesic

I'm learning the book on Information geometry by Amari and Nagoaka after having taken a first course on differential geometry. My question is concerning a geodesic by the $\nabla^{(e)}$-connection. ...
1
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0answers
53 views

Gradient Ricci soliton

I am reading Cao and Chen's paper "On Bach-flat gradient shrinking Ricci solitons". A complete Riemannian manifold $(M^n,g_{ij})$ is called a gradient shrinking Ricci soliton if there exists a smooth ...
2
votes
1answer
40 views

measure on non-oriented Riemannian manifold

Let $M$ be a non-oriented Riemannian manifold of dimension $m$. Nash embedding theorem implies that there exists an isometric embedding $\phi: M\longrightarrow \mathbb{R}^n$ for $n$ sufficiently ...
1
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1answer
34 views

Autoparallel submanifolds and geodesics

I have the following question in differential geometry. Any help is greatly appreciated. Let $M$ be an autoparallel submanifold of a manifold $S$ with respect to a connection $\nabla$. Let $\gamma$ be ...
3
votes
2answers
38 views

How do geodesics change when I scale the metric?

If (M,g) is a Riemannian manifold, and f(m) is a positive real-valued function on M, then f.g is another Riemannian metric on M. If I know all the g-geodesics from x to y in M, can I find out the ...
1
vote
2answers
121 views

Why do people stick with Riemann-Integration when dealing with differential geometry?

I asked a question yesterday that is, "Is there an introductory differential geometry text using Lebesgue integration?" Then, i got an answer that "since we are dealing with differential geometry we ...
0
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0answers
51 views

is there an introductory differential geometry text using Lebesgue integration?

Is there an introductory differential geometry text using Lebesgue integration? Every differential geometry text I saw introduces the theory using Riemann integration. (Even Spivak) Would someone ...
1
vote
1answer
65 views

Metric and Curvature on a Riemann Surface

We are given a smooth conformal metric $\rho=\rho(z)\left|dz\right|$ on a Riemann surface $X$. I have a few questions relating to this: (a) The local formula $R(\rho)=\Delta \mathrm{log}\rho dx\,dy$ ...
2
votes
0answers
69 views

Prove the Curvature Tensor is a Tensor

For an affine connection $\nabla$, prove the curvature R $R(X,Y,Z,\alpha)=\alpha(\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z -\nabla_{[X,Y]}Z)$ with $X,Y,Z$ vector fields and $\alpha$ a co-vector, is ...
10
votes
1answer
102 views

How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
8
votes
1answer
81 views

Using index notation to write $d^2=0$ in terms of a torsion free connection.

Let $(M,g)$ be a Riemannian manifold and let $\omega$ be a $1$-form on $M$. I want to rewrite $d^2\omega=0$ in terms of the Levi-Civita connection. I can show the following: $$d\omega(X,Y) = ...
3
votes
0answers
88 views

$\operatorname{div}$ and $\operatorname{grad}$ in spherical coordinates. Formula from general relativity goes crazy

I try to calculate the gradient of a function and the divergence of a vector field in spherical coordinates. Nothing special so far, but a formula that I learned in a general relativity lecture ...
1
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1answer
43 views

About Whitney Theorem

Note that $M$ of dimension $n$ can be imbedded differentiably as a closed submanifold of ${\bf R}^{N=2n+1}$. Here Let $f$ be an imbedding. $f$ is one-to-one immersion, that is, rank $n$, which is ...
3
votes
1answer
58 views

Flatness of a manifold (or a connection)

Suppose we have an $n$-dimensional manifold $S$ (with a global coordinate system) with a metric $g$ and a connection $\nabla$ with connection coefficients (Christoffel symbols) $\Gamma_{i,j}^k$ given. ...
4
votes
1answer
55 views

surface measure under induced surface metric

I'm currently reading a paper about incompressible Euler's equation, and I don't understand how the surface element expand. So here comes the question. Let $\Omega$ be a Riemannian manifold with ...
0
votes
0answers
21 views

hamiltonian mechanics

In $\mathbb{R}^{2n}$, $\omega=\sum dx_i \wedge dy_i$ is a canonical symplectic form, and H is an hamiltonian function, i.e. $\dot{x}= \frac{\partial H}{\partial y}$, $\dot{y}= -\frac{\partial ...
0
votes
1answer
42 views

The distance in Riemann manifold

Let $f: M\to M$, where $(M,\rho)$ is a closed Riemann manifold, and $(\widetilde{M},\widetilde{\rho})$ is the universal covering of $(M,\rho)$, $D$ is a fundamental domain of ...
2
votes
1answer
58 views

Notation and hierarchy of cartesian spaces, euclidean spaces, riemannian spaces and manifolds

I am confused by some definitions. Forgive the looseness of my language. A Cartesian space is basically a space of points that can be represented by n-tuples ( and other things, but I won't go into ...
1
vote
0answers
35 views

Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
0
votes
2answers
44 views

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 ...
1
vote
2answers
42 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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0answers
53 views

Isometry group of a compact manifold

Is an isometry group of a compact manifold always a compact group?
1
vote
1answer
52 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 ...
0
votes
1answer
34 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?
2
votes
1answer
97 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, ...
1
vote
1answer
39 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 ...
0
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0answers
34 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 ...
3
votes
1answer
53 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 $$ \gamma \, : \, t \, \longmapsto \, \mathrm{Exp}_{p}(tv+v_{0}) $$ ...