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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Unit length tangent vectors on a Riemannian manifold

Let $X$ be a Riemannian manifold and $TX$ its tangent bundle. Is there a name for the $S^1$-bundle given by the unit length tangent vectors?
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371 views

The divergence of the Weyl tensor

First, the Weyl tensor is given by $$W_{ijkl}=R_{ijkl}-\frac{1}{n-2}(g_{ik}A_{jl}-g_{il}A_{jk}-g_{jk}A_{il}+g_{jl}A_{ik})$$ where, $A_{ij}$ is the Schouten tensor, given by ...
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122 views

Visualize $\mathbb{S}^3/\Gamma$!

I thought the only 3-manifold with positive constant curvature is $\mathbb{S}^3$. But I faced $\mathbb{S}^3/\Gamma$, where $\Gamma$ is a finite subgroup of $SO(4)$ and surprised! My problem is that I ...
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236 views

Double of Riemannian manifold.

Let $M$ be a Riemannian manifold with totally geodesic boundary $\partial M$. We consider its double $\check{M}$, i.e. the disjoint union of $M$ with itself under identification of corresponding ...
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Comparation of values in minimal submanifold

Let $\phi: M^m\to H^n(k)$ be minimal immersion. Show that ...
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77 views

Geodesic complete subset of a connected manifold

This may be a very silly question but let us consider a connected Riemanian manifold $(M,g)$ and a subset $O\subset M$. Can we have $O$ geodesic complete (in the sense of all geodesics linking two ...
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114 views

Showing the 3D Ricci flow ODE preserves the order of the curvature tensor eigenvalues

The following system of ODEs arises when studying Ricci flow on 3-manifolds: $$ \frac{dm_1}{dt} = m_1^2+m_2m_3 \\ \frac{dm_2}{dt} = m_2^2+m_1m_3 \\ \frac{dm_3}{dt} = m_3^2+m_1m_2 \\ $$ Going back ...
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203 views

Visualization of the diffeomorphism!

Basic to all mathematics is the notion-here used quite informally-of a set with structure. For every type of structure there is a notion of equivalence (or isomorphism)-a one-to-one onto ...
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143 views

On the ineptitude of the Lie derivative to define directional derivatives

I'm stuck at question 4-3. from John Lee's Riemannian Manifolds: There exists a vector field on $\Bbb R^2$ that vanishes along the $x_1$-axis, but whose Lie derivative with respect to $∂_1$ ...
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180 views

Extending a map of manifolds continuously

Let $M$ and $N$ be manifolds, and $A \subset M$ compact. Let $f:A \rightarrow N$ be a continuous mapping. Show there exists an open neighborhood $U$ containing $A$ and continuous extension $g:U ...
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41 views

Reference request for studying on space forms

I would like to study on Space form, But I dont know what book or notes are suitable for beginning basically. Can someone help me? Thanks.
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47 views

Derivative of tangent vetor

Let $x(\theta,\phi)=(\cos(\theta)\sin(\phi),\sin(\theta)\sin(\phi),\cos(\phi))$ be the standard local coordinates for $\mathbb{S}^{2} \setminus \lbrace (0,0,\pm 1) \rbrace$. Let $\phi_{0} \in ...
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69 views

On the canonical neighborhoods

Suppose $M$ is a 3-dimensional manifold, John W. Morgan and Frederick Tsz-Ho Fong in their "Ricci Flow and Geometrization of 3-Manifolds" book as a definition of canonical neighborhoods have ...
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366 views

Why is arc length not a differential form?

I read that the arc length is not a differential form. But I don't understand why it isn't. I understand that differential forms are integrands and arc length is an expression which is integrable. ...
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1answer
46 views

Formula for curvature of two intersecting surfaces in terms of their normal curvature.

I have been privately reading DoCarmo recently, and have been attempting to do some of the problems. I am stuck on this one, it is problem 14 in section 3.2 for those interested. If someone could show ...
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581 views

Exponential map on the the n-sphere

I might need some help on the following exercise : Let $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$ be the unit $n$-sphere. For any $p \in \mathbb{S}^{n}$, we have $T_{p}\mathbb{S}^{n} = p^{\perp} = ...
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57 views

Connected sum while keeping curvature bounded.

Is it possible to perform a connected sum of two Riemannian Manifolds or Orbifolds while keeping curvature bounded from below? More explicitly, If $M_1$ and $M_2$ are two Riemannian manifolds (or ...
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80 views

Tensors: summing over indices

Would anybody mind teaching me how to work these indices? Definitions: Throughout the following, repeated indices are to be summed over. Hodge dual of a p-form $X$: $$(*X)_{a_1...a_{n-p}}\equiv ...
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67 views

A positive constant don't change the Levi-Civita Connection

In the Chow's book there is a question that I can't solve, the question is Let $\nabla^g$ denote the Levi-Civita connection of the metric $g$. Show that for any constant $c>0$ and metric $g$, ...
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101 views

Riemannian Distance function taylor series

Given a Riemannian manifold $ (M,g) $, in a neighborhood of a fixed point $ x $, the distance function $ d^2(x,\cdot) $ is smooth. In coordinates, the taylor series starts off like: $$ d^2(x,y) = ...
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553 views

Metric of the flat torus

I am studying the flat torus $T^n=\mathbb{R}^n/\mathbb{Z}^n$. I am interested in the metric and the connection used. Unfortunately, in the books I am reading those things aren't defined. Does anyone ...
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30 views

Rotationally symmetric hypersurfaces with mean curvature bounded away from 0

I know that the rotationally symmetric hypersurfaces in $\mathbb{R}^n$ with constant mean curvature are the hyperplane, sphere, cylinder, catenoid, nodoid, and unduloid. Are there any significant ...
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106 views

When does a Green's function exist?

If I have a simply-connected compact domain $ \Omega $ in $ \mathbb{R}^2 $, endowed with a Riemannian metric $ g $, does there exist a green's function on $ \Omega $ for the laplace operator induced ...
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66 views

Automorphism on a kahlerian variety which acts as -id on $H^2(X,\mathbb{Z})$

I've read this proposition: "there is no automorphism (biholomorphic map) of a Kahlerian complex manifold $X$ which acts on $H^2(X,\mathbb{Z})$ as $-$identity" so I tried to prove this statement and ...
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Isotropic Manifolds

Recall that a Riemannian manifold $(M,g)$ is isotropic if for any $p\in M$ and any unit vectors $v,w\in T_pM$ there is an isometry $f:M\to M$ such that $f_\ast(v)=w.$ Recall also that $(M,g)$ is ...
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108 views

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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82 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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63 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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150 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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74 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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111 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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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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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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212 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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188 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 $M^n$ ...
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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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189 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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76 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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64 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.
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159 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 ...
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443 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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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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90 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 ...
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59 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 ...
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215 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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72 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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94 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 ...
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113 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 ...
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36 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 ...
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114 views

Second variation formula and Jacobi fields

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