(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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179 views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
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1answer
54 views

Definition of the integral of a vector field on Riemannian manifold and Euclidean spaces

Given a compact Riemannian manifold $(M,g)$ and a vector field $X \in \mathfrak{X}(M)$, is it possible to define the integral of $X$ on $M$? What if $M$ is a Euclidean space? Clearly the definition ...
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0answers
101 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
2
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2answers
44 views

Are all critical points of energy geodesics?

Let $\gamma$ be a smooth curve in a Riemannian manifold and consider the arclength functional $L(\gamma) = \int_a^b |\gamma'(t)|\, dt$ and the energy functional $E(\gamma) = \frac{1}{2}\int_a^b ...
2
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0answers
36 views

Why Riemannian metrics defining the same angles are conformal [duplicate]

Suppose $g_1$ and $g_2$ are two metrics defining the same angles, which means $g_1(X,Y)/(g_1(X,X)g_1(Y,Y))^{0.5}=g_2(X,Y)/(g_2(X,X)g_2(Y,Y))^{0.5}$ for all pairs of vector $X,Y$.I want to prove that ...
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0answers
60 views

Completeness of a Riemannian manifold with boundary

I have some issues understanding the notion of completeness of a Riemannian manifold with boundary. In the case of Riemannian manifolds without boundary, I found that completeness is usually defined ...
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1answer
52 views

Request for online reference to Hamilton's “The Ricci Flow on Surfaces”

Does anyone know of an online source for Richard Hamilton's paper "The Ricci Flow on Surfaces?" I've searched Google for it and it doesn't seem to give any results.
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1answer
129 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}+ ...
4
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1answer
78 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 ...
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1answer
16 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 ...
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0answers
25 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
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1answer
65 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
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1answer
57 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 ...
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1answer
33 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
74 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
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1answer
81 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 $M$ ...
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1answer
62 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 ...
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0answers
24 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 ...
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0answers
28 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 ...
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0answers
27 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 ...
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1answer
35 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 ...
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0answers
40 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 ...
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1answer
39 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 ...
2
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1answer
80 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 ...
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2answers
61 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 ...
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0answers
50 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 $n=1$, ...
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0answers
162 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 ...
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0answers
42 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 ...
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2answers
62 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 $\Omega(X)$ ...
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0answers
35 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. ...
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0answers
70 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 ...
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1answer
58 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 ...
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1answer
63 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 ...
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2answers
51 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 ...
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2answers
130 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 ...
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0answers
52 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 ...
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1answer
98 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 $R(\rho)=\Delta \mathrm{log}\rho dx\,dy$ ...
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0answers
91 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 ...
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1answer
119 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 ...
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votes
1answer
92 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) = ...
4
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0answers
105 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 ...
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1answer
45 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 ...
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1answer
73 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. ...
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1answer
70 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 ...
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1answer
35 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 ...
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1answer
43 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
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1answer
78 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 ...
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0answers
36 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 ...
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2answers
47 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 ...
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2answers
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 ...