(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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Seifert surfaces in Riemannian manifolds?

Does there exist an equivalent to Seifert surfaces for other Riemannian manifolds than $\mathbb{R}^3$? More precisely: Let $M$ be a simply-connected Riemannian manifolds and $K \subset M$ a (tame) ...
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44 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
69 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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1answer
31 views

Does a simply connected complete riemannian manifold with POSITIVE upper curvature bound have positive injectivity radius?

For example: I am thinking that some sort of rauch comparison theorem could be helpful
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1answer
12 views

Time evolution of Laplacian

While reading monograph on the Ricci flow, I came accross a fact (at least I think it is a fact), which is not proved explicitly in that book. Assume a smooth 1-parameter family of Riemannian metrics ...
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2answers
102 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
52 views

Question about Hopf-Rinow theorem

I'm studying Hopf-Rinow theorem and I don't see a step in the proof. Could someone help me, please? (Definition) Let's $(M, \langle,\rangle)$ an ANII(axiom numerability 2) and Hausdorff Riemannian ...
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3answers
85 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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9 views

The time evolution of levi-Civita connection

Assume a smooth one-parameter family of Riemannian metrics $g_{t}$. Write $h:=\frac {\partial}{\partial t}g$. In addition, assume that the Levi-civita connection on the Riemannian manifold ...
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1answer
33 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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73 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 ...
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32 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 ...
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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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1answer
39 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
51 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
93 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}+ ...
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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 ...
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1answer
49 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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0answers
18 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 $\mathbb{S}^n_+$, ...
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1answer
232 views

Riemannian measure and Hausdorff measure in a general Riemannian Manifold

Let $ M $ be a Riemannian manifold and let $ \mu $ be its Riemannian measure. This is the measure obtained by Riesz reprersentation theorem such that for every continuous function with compact support ...
4
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1answer
40 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 + ...
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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 ...
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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 ...
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1answer
68 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
57 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
25 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
46 views

Sobolev diffeomorphisms.

Let $M$ be a compact Riemannian manifold without boundary. Suppose $f \in H^s(M,M)$, where $H^s$ denotes the ($L^2$-based) Sobolev space. Assume $s > n/2 + 1$, so that by the Sobolev Embedding ...
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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 ...
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0answers
18 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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21 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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149 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
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 ...
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1answer
34 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 ...
5
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2answers
364 views

Geodesics on the product of manifolds

Given two Riemannian manifolds $(M, g_1)$ and $(N, g_2)$, and geodesic curves $\gamma(t)$ in $M$ and $\chi(t)$ in $N$. Is the curve $\Gamma(t) = (\gamma(t),\chi(t))$ a geodesic in the product manifold ...
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2answers
55 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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47 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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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 ...
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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 $\Omega(X)$ ...
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0answers
31 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
55 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
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1answer
43 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
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 ...
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2answers
40 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
122 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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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 ...
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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 ...
2
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0answers
71 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
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 $R(\rho)=\Delta \mathrm{log}\rho dx\,dy$ ...
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103 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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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) = ...