(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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An Exercise in Peter Petersen

My question is an exercise in Peter Petersen "Riemannian Geometry" Chapter 5 #10 Let $N \subset M$ be a submanifold of Riemannian manifold $(M,g)$. (a) The distance from N to $x \in M$ is defined as ...
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57 views

A little question about equidistants of a embedded hypersurface

Assume M is an oriented manifold and $W \subset M$ is a smooth compact embedded hypersurface without boundary. Is there an example that the outer equidistants $W_t$ is not smooth for all $t$ in some ...
3
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395 views

Commutation formula for covariant derivative

Suppose $\nabla$ is the Levi-Civita connection on Riemannian manifold $M$. $X$ be a vector fields on $M$ defined by $X=\nabla r$ where $r$ is the distance function to a fixed point in $M$. $\{e_1, ...
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316 views

Prove that Killing vector fields form Lie algebra.

I want to find the integral curves of $[X,Y]$, then maybe can use this to prove. Can anyone gives an answer ? Thanks.
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178 views

A smooth function f satisfies $\left|\operatorname{ grad}\ f \right|=1$ ,then the integral curves of $\operatorname{grad}\ f$ are geodesics

$M$ is riemannian manifold, if a smooth function $f$ satisfies $\left| \operatorname{grad}\ f \right|=1,$ then prove the integral curves of $\operatorname{grad}\ f$ are geodesics.
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362 views

Approximate expression for the metric in normal coordinates

In the Wikipedia article on Ricci curvature (here) it is mentioned that one can approximate the metric g in normal coordinates by \begin{equation} g_{ij} = \delta_{ij} - \frac{1}{3} R_{ikjl} \,x^kx^l ...
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75 views

Simplifying the search for a geodesic

How can calculations for the geodesics on the surface $U=\{(x,y,z): c(x^2+y^2)-z^2=0, z>0\}$ be simplified by noting that is locally Euclidean? I can see that the property means that when we open ...
3
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295 views

Thrice-punctured sphere

This claim is made in the book Quantum Triangulations (eds.: Carfora, Marzuoli), p.45: the thrice-punctured sphere is the largest subdomain of $\mathbb{S}^2$ supporting a hyperbolic metric. I ...
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122 views

Riemannian metric - basic question

I tried to google the following but couldn't find an answer that helped - so I hope I might find some here - the question is short and very basic (I guess) : what does it mean when someone writes ...
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332 views

Adjoint of the covariant derivative on a Riemannian manifold

Let $\nabla_X$ be the covariant derivative on a Riemannian manifold w.r.t. the vector field $X$. It is not clear to me what the (formal) adjoint of this operator is: I mean the operator ...
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372 views

How to go from local to global isometry

Let $M$ be a connected complete Riemannian manifold, $N$ a connected Riemannian manifold and $f:M \to N$ a differentiable mapping that is locally an isometry. Assume that any two points of $N$ can be ...
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109 views

Yet another natural Connection on Riemannian manifolds?

The exterior derivative $d:\mathcal{A}^1(M)\to\mathcal{A}^2(M)$ can be regarded as an connection on $T^*M\to M$. If $g$ is a Riemannian connection on $M$, we can can pull $d$ back to get an connection ...
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82 views

Orthonormed vector fields on a Riemaniann surface

Let $S$ be a $\mathcal{C}^{\infty}$Riemannian surface. Consider $x \in S$. Can I always find two smooth vector fields $X$,$Y$ defined in a neighborhood $V$ of $x$ such that $\forall y \in V$ ...
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1answer
88 views

Complexification of Metric

If we have given an inner product space $(V,g)$, where $V$ is vector space and $g$ is inner product. What will be corresponding bi-linear form $g'$ on $C\otimes V$.
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114 views

Help on Einstein Summation

I am not sure how to interpret the following expression with regard to the Einstein summation convention \begin{equation} g^{ab}(\partial_c \Gamma^c_{ab} - \partial_b \Gamma^c_{ac}) \end{equation} ...
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126 views

What's the relationship between the riemannian metric and Jacobi field?

I encounter to the question in reading the following Excise: Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ be the (geodesic) polar ...
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382 views

Can every curve on a Riemannian manifold be interpreted as a geodesic of a given metric?

Given a metric $g_{\mu\nu}$ it is possible to find the equations of the geodesic on the Riemannian manifold $M$ defined by the metric itself: $$\frac{d^2x^a}{ds^2} + ...
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Existence of a local geodesic frame

Let $(M,g)$ be a Riemannian manifold of dimension $n$ with Riemannian connection $\nabla,$ and let $p \in M.$ Show that there exists a neighborhood $U \subset M$ of $p$ and $n$ (smooth) vector fields ...
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264 views

antipodal map of complex projective space

Let $CP(n)$ be the complex projective space with Fubini-Study metric with diameter $=\frac{\pi}{2}$. Fix a point say $p\in CP(n)$; my question is what is the set of points of maximum distance to the ...
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860 views

Isometries preserve geodesics

Let $f$ be an isometry (i.e a diffeomorphism which preserves the Riemannian metrics) between Riemannian manifolds $(M,g)$ and $(N,h).$ One can argue that $f$ also preserves the induced metrics $d_1, ...
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107 views

Coordinate-free proof of the hamiltonian character of the geodesic flow

Let be $(M,g)$ a pseudoriemannian manifold. Let us identify the tangent and the cotangent bundles through the musical isomorphism $g^\flat:u\in TM\to g(u,\cdot)\in T^\ast M.$ It is well known ...
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234 views

the hyperbolic plane is complete

I'd like to prove that the hyperbolic plane is complete in a short, nice way. Here's my proof, but I'm not convinced of its correctness yet. Let $\mathbb{H}=\left\{(x,y)\in\mathbb{R}^2\big\vert ...
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317 views

Existence of a Riemannian metric inducing a given distance.

Let $M$ be a smooth, finite-dimensional manifold. Suppose $M$ is also a metric space, with a given distance function $d: M \times M \rightarrow \mathbb{R}_{+}$, which is compatible with the original ...
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182 views

The set of diffeomorphisms preserving some metric.

Let $M$ be a finite-dimensional, smooth manifold. Call a diffeomorphism $f : M \rightarrow M$ diagonalizable if there exists a Riemannian metric $g$ on $M$ such that $f : (M, g) \rightarrow (M, g)$ is ...
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isometry group of the riemannian manifold $\mathbb{T}^2$?

What is the isometry group of the riemannian manifold $\mathbb{T}\times \mathbb{T}$ where $\mathbb{T}=\{z\in \mathbb{C}\ :\ |z|=1\}$ is the classical torus?
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Interpreting the scalar curvature in a semi-Riemannian manifold

Background: Let $M$ be a smooth Riemannian manifold of dimension $n$ and scalar curvature $R$ (with respect to the Levi-Civita connection). Let $m \in M$ and let $B$ be the geodesic ball of radius ...
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164 views

What complete local invariants are there in Riemannian geometry? [closed]

What complete local invariants are there in Riemannian geometry? Specifically, is Riemannian curvature tensor such a complete local invariant?
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229 views

scalar curvature on one - dimensional Riemannian Manifold

How can i express the scalar curvature for a one - dimensional Riemannian manifold (M, g) in terms of the metric g ?
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The space of Riemannian metrics on a given manifold.

For a finite-dimensional smooth (Hausdorff, second-countable) manifold $M$, consider the set $$\mathcal{Met}(M) = \{ g : g \text{ is a Riemannian metric on }M \}.$$ I'd like to know about the typical ...
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917 views

Conformal transformation of the curvature and related quantities

Suppose we have a Riemannian manifold ${(M,g)}$, where ${g}$ is the metric of ${M}$. If ${f}$ ${\in}$ ${D(M)}$ (i.e. smooth function on ${M}$), and ${f}$ is positive. So, we can define a new metric ...
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959 views

Problems in do Carmo's Riemannian Geometry

I am reading do Carmo's Riemannian Geometry Chapter 7, and I want to do some exercises. I think that I need some hints to solve the following: Questions: How do I construct a counterexample that a ...
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246 views

Questions about Hyperbolic Isometries: The Standard Inversion

I have two questions regarding the inversion across the unit circle in the hyperbolic plane. Recall that the hyperbolic plane is a metric space consisting of the open half-plane $$\mathbb{H}^2 = ...
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Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal iff $g$ is flat

Let $(M,g)$ be a Riemannian manifold. Then I want to show that these are equivalent: (i) Each point of $M$ has a smooth coordinate neighborhood in which the coordinate frame is orthonormal. ...
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1answer
515 views

Alternate definition of a 'geodesic ball'

Background: Let $M$ be a Riemannian manifold. Let $p \in M$ and $\epsilon \gt 0$. For sufficiently small $\epsilon$, the standard definition (correct me if I'm wrong) for the 'geodesic ball of ...
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246 views

metric on the universal covering of a geometric manifold

We know that the universal covering of a closed hyperbolic 3-manifold can be identified with the hyperbolic space $\mathbb{H}^3$. Now, what is not very clear to me is how this identification has to be ...
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1answer
285 views

exponential map and sectional curvature

Let $\Pi$ be a nondegenerate tangent plane to $M$, a semi-Riemannian manifold, at $p$. If $P$ is a small enough neighborhood of 0 in $\Pi$. What is the Gaussian curvature at $p$ of $\exp_p(P)$?
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1answer
160 views

Simple problem with the normal curvature tensor

If $M$ is a s-R(semi-Riemannian) submanifold of a s-R manifold $\overline{M}$ the function $R^{\perp}:\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M)^\perp\rightarrow\mathfrak{X}(M)^\perp$ ...
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1answer
158 views

A semi-Riemannian geometry exercise

How can I prove that there are no compact semi-Riemannian hypersurfaces in semi-euclidean space $\mathbb{R}_v^n$ of index $v$ with $0<v<n$??. Thanks for any help!!
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What is the intuitive meaning of the scalar curvature R?

Background: Let $M$ be a smooth, Riemannian manifold with metric $g$ and dimension $n$. Let $R^a_{bcd}$ be the Riemann tensor with respect to the Levi-Civita connection for $g$. Question: Is there ...
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1answer
447 views

Killing vector fields restricted to geodesics

Given a Riemannian manifolds $(M,g)$, a Killing vector field $X$ on $M$, and a geodesic $\gamma: K \rightarrow M$ defined on an interval $K \subseteq \mathbb{R}$, how does one show that $X \circ ...
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Any compact embedded $2$-dimensional hypersurface in $\mathbb R^3$ has a point of positive Gaussian curvature

Problem statement: Let $M \subseteq \mathbb{R}^3$ be a compact, embedded, 2-dimensional Riemannian submanifold. Show that $M$ cannot have $K \leq 0$ everywhere, where $K$ stands for the Gauss ...
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1answer
92 views

Is $\Gamma_{jk}^{t}=\frac{1}{2}g^{it}g_{ij,k}$?

Here $\Gamma_{jk}^{t}$ is the Christoffel symbol of the second kind, and $g$ is the Riemann metric on a Riemann manifold.When learning Riemann Geometry, we are usually introduced to the following ...
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1answer
133 views

Backslash notation: $\Gamma {\setminus} \mathbb{H}^n$

I encountered this notation in a paper by Carron: When X = $\Gamma{\setminus}\mathbb{H}^n$ is a real hyperbolic manifold, ... $\Gamma$ is a discrete torsion free subgroup of SO$(n,1)$. My ...
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232 views

Question in do Carmo

In Exercise 9, Chaper 4, do Carmo gives a hint to solving a problem. He says to consider an orthonormal basis $e_1, \ldots, e_n$ in $T_pM$ such that if $x = \sum_{i=1}^n x_i e_i$, $$\text{Ric}_p(x) = ...
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“Maximal symmetry” metric for a manifold?

So this is my question : Let $M$ be a smooth manifold. With any riemaniann metric $g$ on $M$ comes an isometry group $I(g)$. Intuition (well at least mine, which may be flawed...) suggests that ...
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892 views

Proof of Bochner formula/ Weitzenböck formula in a non-normal frame

The proof of the classical Weitzenböck formula $$ \Delta (|f|^2)=|{\rm Hess}f|^2+\langle\nabla f, \nabla (\Delta f) +{\rm Ric} (\nabla f, \nabla f) \rangle $$ uses the local orthonormal ...
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Definitions of Hessian in Riemannian Geometry

I am wondering is there any quick way to see the following two definitions of Hessian are coinside with each othere without using local coordinates? $\operatorname{Hess}(f)(X,Y)= \langle \nabla_X ...
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The exponential map

I'm following a course about riemannian geometry, and I was fascinated with the exponential map. I was wondering what the reason of this name is... is there any relationship with the real and complex ...
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1answer
148 views

Deformation of the Kähler structure on $CP^n$

Using the homogeneous coordinate on $CP^n$, we consider the open set $U_0:=\{[1, \ldots, z_n]\}$. Then the standard Kähler form of $CP^n$ can be written as $$ ...
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Question about the proof of the index theorem appearing in Milnor's Morse Theory

I am trying to get through the proof of the index theorem. The background: I have been stuck for quite a while on the following point which Milnor says is evident: Let $\gamma: [0,1]\rightarrow M$ be ...