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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why positive scalar curvature manifolds

I am studying scalar curvature and I have seen that many mathematicians studied obstruction against positive scalar curvature (for example Stolz, Schick, Roe, J. Rosenberg, Hanke and many others). ...
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87 views

scalar curvature

I am studying scalar curvature. It is the trace of the Ricci operator. I read that its geometric meaning follows from this formula ...
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1answer
64 views

Covariant derivative of $Ricc^2$

How to derivate (covariant derivative) the expressions $R\cdot Ric$ and $Ric^2$ where $Ric^2$ means $Ric \circ Ric$? Here, $Ric$ is the Ricci tensor seen as a operator and $R$ is the scalar curvature ...
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184 views

Approximate parallel transport using Jacobi fields

Let $M$ be a riemannian manifold (let $\left\langle \cdot,\cdot \right\rangle_{p}$ be the scalar product on $T_{p}M$). Let $p \in M$ and $\xi \in T_{p}M$. We consider the geodesic $\gamma \, : \, t \, ...
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76 views

Construct harmonic function on noncompact manifold

$M$ is a non-compact Riemannian manifold, $p \in M$. Consider Dirichlet problems: $\Delta u = 0$ in ${B_p}\left( i \right)$ ($i = 1,2, \dots $), $u{|_{\partial {B_p}\left( i \right)}} = {f_i}$, ${f_i} ...
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119 views

Torsion of fundamental group is abelian

On Riemannian manifold with Ricci curvature bounded below (For example, flat torus. It has ${\bf Z}^2$ as fundamental group, which is nilpotent (=almost abelian). In fact Ricci curvature bouned ...
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121 views

Smooth partitions of unity

Let $ M $ be a Riemannian manifold and let $ \{U_i\} $ be a countable covering of $ M $. It is well known that there exists a countable collection of smooth function with compact support $ \{\rho_i\} ...
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82 views

Notation in riemannian geometry

I am reading a lecture on Riemannian geometry in which it is written that, for a differentiable manifold $M$ and a differentiable curve $v \, : \, I \, \longrightarrow \, M$ defined on an interval ...
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226 views

Different Riemannian manifolds with the same Riemannian volume form

Let $(X,g_1)$ and $(X,g_2)$ be two Riemannian manifolds over the same space $X$. My (vague) question is the following : If I know that the two induced Riemannian volume form coincide, what can I say ...
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1answer
214 views

Rotationally symmetric metrics and Jacobi Equation

I've the following doubt: Let $(M^n,g)$ a Riemannian Manifold, where $g=dr^2+\gamma^2(r)d\omega^2$ is given in geodesic spherical coordinates. Suppose that the radial sectional curvatures of $M^n$ ...
3
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80 views

Components of the Riemann tensor

Here is a short question which has been bugging me for a long time: In many textbooks, the components of the Riemann curvature in local coordinates/abstract index notation are defined as follows: If ...
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214 views

About Hessian of distance function

I'm studying the comparison Hessian theorem and I not understand the following: Let $(M, \langle\ ,\ \rangle)$ be a complete Riemannian manifold. Given $o\in M$, define $r=dist(o, \cdot)$. Then, for ...
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91 views

pre-quantization on cotangent bundle $T^*M$

Let M be a manifold. Is there any pre-quantization on cotangent bundle $T^*M$?. In which case this pre-quantization is unique?
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1answer
66 views

Definition question of convex orbit of finite group action

Assume that a finite group or discrete group $G$ acts on a manifold $M$. Here what does it mean that orbit $G\cdot x$ is convex ? Thank you in advance.
2
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1answer
164 views

The pointed Gromov-Hausdorff limit of action on manifold

The pointed Gromov - Hausdorff limit is a concept of the convergence of Riemannian manifolds : For instance $$ (\lambda_i S^2(1) , p) \rightarrow_{G-H} ({\bf R}^2,O) = T_p S^2(1)$$ where ...
6
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489 views

weak solutions versus classical solutions

Let $ \Omega $ be an open subset with compact closure of a Riemannian manifold $ M $. Let $ u \in H^1_{0}(\Omega) $ be a weak solution of the Dirichlet boundary problem: $$ -\Delta u + qu = f \; \; ...
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3answers
146 views

Comparison Statement

I'm looking for a statement like this (and for a proof too): Let $\gamma_1,\gamma_2:\mathbb R\to (M,g)$ two curves (parametrized by arclength) in a Riemannian manifold $(M,g)$ and let ...
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1answer
93 views

If $M$ is complete is the closed ball compact?

Let $M$ be a Riemannian manifold and $p,q \in M$. Let $\Omega=\Omega(M;p,q)$ be the set of piecewise $C^\infty$ paths from $p$ to $q$. Let $\rho$ denote the topological metric on $M$ coming from its ...
3
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1answer
61 views

Tangent space of loop space.

Let $\Omega$ be the path space of a riemannian manifold $M$. I have to define the tangent space of $\Omega$ in a path $\omega$, that I denote with $T_p \Omega$. I think that this space is the vector ...
3
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2answers
235 views

Minimizing energy functional

Let $ \Omega $ be an open subset with compact closure and smooth boundary of a non compact riemannian manifold $ M $. Let $ f \in C^{\infty}(\partial \Omega) $ and $ q \in C^{\infty}(M) $, $ q \geq 0 ...
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0answers
77 views

isomorphism of two compex line bundles

I am looking for some non-trivial examples of Line Bundles and an example about isomorphism of two line bundles. With details
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0answers
96 views

Parallel transport on a submanifold

Let $M$ be a Riemannian manifold and $N$ an embedded submanifold of $M$. Now I have a geodesic $c \colon (-a,a) \to N$ with $a>0$ and $c(0)=p$ in $N$, s.t. $c'(0)$ is orthogonal to $T_pN$. Is the ...
2
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1answer
143 views

Gaussian connection on submanifold in Euclidean space: how to compute Christoffel coefficients

Consider an embedding $$\boldsymbol{r}=\boldsymbol{r}(u^1,\ldots,u^n)$$ of some $n$-dimensional manifold $M$ in $\mathbb{R}^N,$ with the correspondind induced metric. We know that both ...
2
votes
2answers
38 views

Manifolds with finitely many ends

In the article ' The structure of stable minimal hypersurfaces in $ R^{n+1} $ ( http://arxiv.org/pdf/dg-ga/9709001.pdf) of Cao-Shen-Zhu the remark 2 at page 3 contains a statement that i don't ...
2
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1answer
135 views

Second variation formula and Jacobi fields

Let $\bar{\alpha}:U\rightarrow \Omega$ be a two-parameter variation of a geodesic $\gamma$. For $i=1,2$ we define $$W_{i}=\frac{\partial \bar{\alpha}}{\partial u_{i}} \in T_{\gamma}\Omega$$ the ...
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24 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
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2answers
42 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
2
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1answer
75 views

Hamiltonian reduction in symplectic geometry

If $V$ is symplectic and $W^\perp \subseteq W\subseteq V$, then why is $W$ a pre-symplectic vector space? Why is $W/W^\perp$ symplectic?
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1answer
62 views

Path spaces and induced maps on tangent spaces

Let $M$ be a smooth manifold and $\Omega(M)$ the set of all piecewise smooth path in $M$. Let be $$ f: \Omega(M) \rightarrow \mathbb{R} $$ How can I define $$ f^*: T_{\omega}\Omega \rightarrow ...
2
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0answers
83 views

angle sum for triangle on helicoid

Given the helicoid $$ \boldsymbol{r} = (u\sin v, u\cos v,v)$$ in three-dimensional Euclidean space, consider the triangle $T$ defined by $$ 0 \leq u \leq \sinh v, \qquad 0 \leq v \leq v_0.$$ The ...
4
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1answer
360 views

Berger's theorem on holonomy

Can someone clarify to me what the correct hypothesis of Berger's theorem are (if at all what I write is correct)? Theorem: assume $M$ is a Riemannian manifold, with irreducible reduced holonomy ...
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1answer
90 views

projection onto the nullspace of the Laplacian on a conformally compact surface

Let $(M,g)$ be a conformally compact surface. An example situation is a hyperbolic surface of infinite area like the quotient $\Gamma\backslash \mathbb{H}$, where $\mathbb{H}$ is the hyperbolic plane ...
8
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1answer
438 views

Positive definiteness of Fubini-Study metric

Define the Fubini-Study metric $$g_{i\overline{j}} = \frac{\delta_{i\overline{j}}(1+|\boldsymbol{z}|^2)-\overline{z}^jz^i}{(1+|\boldsymbol{z}|^2)^2} $$ for $i,j=1,\ldots,n$ and $z_i$ complex variables ...
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1answer
114 views

conformally euclidean metric on riemannian surface

We know that in euclidean space $\mathbb{R}^3$ the euclidean metric induces on a 2-dimensional surface a riemannian metric which can be brought into conformal form by means of a local change of ...
4
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1answer
272 views

local isometry for riemannian manifolds is not transitive

Let $(M_1,g_1)$ and $(M_2,g_2)$ be Riemannian manifolds of the same dimension, and let $\phi: M_1 \to M_2$ be a smooth map. We say that $\phi$ is a local isometry if $g_2 (\phi_* X, \phi_* Y ) = g_1 ...
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1answer
202 views

Differentiability of the distance function and cut locus

Let $ M $ be a complete Riemannian manifold and let $ d: M \rightarrow R $ be the distance function from a given point $ 0 \in M $. I want to prove that $ d $ is a smooth function on $ M -(C(p)\cup ...
3
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1answer
79 views

How does $\operatorname{Ric} \ge 0$ guarentee the Busemann function is regular in the splitting theorem?

Cheeger-Gromoll's famous splitting theorem says If $(M,g)$ contains a line and $\operatorname{Ric} \ge 0$. Then $(M,g)$ is isometric to a product. I want to know how does $\operatorname{Ric} \ge ...
4
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1answer
207 views

Dimension of isometry group of complete connected Riemannian manifold

Given an $n$-dimensional geodesically complete connected Riemannian manifold $M$, we want to prove that the dimension of its isometry group is $$\dim {\rm ISO}(M) \leq \frac{n(n+1)}2.$$ Does it ...
2
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1answer
58 views

Minimum ratio between surface and volume in a riemannian manifold

In an euclidean three - dimensional space the sphere is the geometric figure with the minimum ratio $R=\frac{S}{V}$ with $S=4\pi r^2$ and $V=\frac{4}{3}\pi r^3$, so we have: $$R=\frac{1}{3}r$$ where ...
4
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2answers
227 views

Eigenvalues of the laplacian on a compact manifold without boundary

Let $ M $ be a compact manifold WITHOUT boundary. It is clear that the first eigenvalue of the Laplace operator $ -\Delta $ is $ \lambda_0=0 $. Now we suppose that M has constant sectional curvature ...
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1answer
104 views

Do carmo problem. exe 6 section 8

Calculate the mean curvature and the sectional curvature of the umbilic hypersurface of the hyperspace. please introduce a book that calculate this. or show how i can calculate this. This is a part ...
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1answer
70 views

Totally geodesic immersions

Let $ x: M \rightarrow \overline{M} $ be a totally geodesic immersion, where $ M $ is a $ k- $ dimensional Riemannian manifold and $ \overline{M} $ is a $ n- $ dimensional Riemannian manifold. Is it ...
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2answers
235 views

Question in do Carmo's book Riemannian geometry section 7

I have a question. Please help me. Assume that $M$ is complete and noncompact, and let $p$ belong to $M$. Show that $M$ contains a ray starting from $p$. $M$ is a riemannian manifold. It is ...
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100 views

Asymptotic invariants of infinite groups

I am reading a Gromov's book " Metric Structures for Riemannian and Non-Riemannian Spaces ". Consider the following concept : $$ distort(X)\doteq sup \frac{length\ dist|_X}{dist|_X} $$ That is for ...
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3answers
187 views

How to define intrinsic curvature?

I have been exploring differential geometry slightly... And i'm trying to grasp how to define intrinsic curvature, from a visual/geometric viewpoint... One formulation I got was that Intrinsic ...
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1answer
77 views

Why $\Sigma$ is minimal, if $\frac{d}{dt} |_{t=0} \mathrm{Area}(\Sigma_t)=0$?

In this work http://arxiv.org/pdf/1204.2883v1.pdf Martin Li claimed that $\Sigma\subset M$ is minimal and $\Sigma$ meets $\partial M$ orthogonally along $\partial \Sigma$ if, only if, $$0 = ...
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28 views

signature of pseudo-Riemannian metric made of Newton polynomials

Given a polynomial with roots $x_1,\ldots,x_n$ and real coefficients, it can be written $$ P(x)=\prod_{i=1}^n \left( x-x_i \right);$$ define Newton polynomials $$s_k(x_1,\ldots,x_n):=\sum_{i=1}^n ...
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3answers
339 views

A question about geodesics of hyperbolic space

Let $ H^n $ be the the upper half space of $ R^n $ endowed with the conformal metric $ g=\frac{1}{x_{n}^{2}}|dz|^2 $ ($ |dz|^2 $ is the standard metric of $ R^n $). This space is the classical ...
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1answer
123 views

Singularity models of the Ricci flow

I faced this sentence in my studies on Ricci flow: The Bryant soliton is a singularity model for the degenerate neckpinch. Q1: What is the definition and meaning of singularity model? Can one model ...
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Are any two smooth Cauchy surfaces of a globally hyperbolic manifold diffeomorphic?

Let $(M,g)$ be a globally hyperbolic Lorentzian manifold. It is known that there exists a smooth Cauchy surface $\Sigma\subset M$ and that $M$ is diffeomorphic to $\mathbb{R}\times\Sigma$. I suspect ...