(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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Upper bound on volume growth

If the Ricci curvature of a compact Riemannian manifold of demsnion $n$ is greater than 1-n, does it follow that the volume entropy satisfies $$\liminf_{r\rightarrow \infty} \frac{\log vol ...
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1answer
159 views

Calculating Geodesics for submanifolds

I am trying to become acquainted with the notion of geodesics. When we consider a Submanifold $M\subset \mathbb{R}^n$ and a curve $c:I\rightarrow M$. Now I want to know how to check whether c is a ...
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1answer
245 views

Some results about Levi-Civita connection in euclidean space?

I calculated some properties of the Levi-Civita Connection on a semi-riemannian manifold. But I'm not sure, whether my results are correct. Can you please tell me, when something below is wrong or if ...
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1answer
127 views

A new metric involving curves

Let $(X, d)$ be a metric space. The inner metric or length metric associated with $d$ is the function $d_i : X \times X \to [0,\infty]$ defined by $$d_i(x, y) := \inf L(\sigma)$$ where the infimum is ...
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108 views

Question about immersions of $\mathbb{R}P^n$ into $\mathbb{R}^{n+1}$

I am currently reading a paper which takes for granted the following geometric fact: if $\mathbb{R}P^n$ can be immersed in $\mathbb{R}^{n+1}$ then for some $k$, $n=2^k-1$ or $n=2^k-2$. My initial ...
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631 views

Isometries of $\mathbb{S}^n$

How to prove so elementary (elementary = without using the concept of geodesic) that an isometry of $\mathbb{S}^n$ is a restriction on $\mathbb{S}^n$ of an isometry of $\mathbb{R}^{n+1}$ ? EDIT: ...
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0answers
178 views

Extending Tensor Fields defined on Manifolds to Ambient Space

I am currently reading about tensor fields on manifolds, and I came across two comments that sound contradictory to me. The first comment is made in the book by James Munkres "Analysis on Manifolds", ...
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1answer
260 views

Proving a curve is a geodesic.

I am really stuck on the following question. Let $ \ \gamma : I \longrightarrow M \ $ be a non-constant (i.e $ \ \gamma'\ $ is not identically zero) geodesic. Show that a reparametrization $\ ...
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485 views

Precise definition of conformal structure based on a Riemannian metric on a Riemann surface

As I read the literature, I keep having some doubt about what a " conformal structure on a Riemann surface " exactly means. ( You can assume all the Riemann surface in this literature have universal ...
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0answers
182 views

Hopf-Rinow theorem

If $(M,g)$ is a riemannian manifold. $M$ is complete(geodesically) then any two point can be joined by a geodesic. Geodesic($\gamma(t))$ is smooth curve such that ...
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1answer
143 views

Non unique solution for Ricci flow equation

Why completeness is important for the uniqueness of solution to Ricci flow? For example, if $M$ is the open unit disk in $\mathbb{R}^2$ and $g(0)$ is the Euclidean metric, and hence not complete. Why ...
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458 views

Cigar soliton solution

In wikipedia, it has been written that an important 2-dimensional example of Ricci flow over $M=\mathbb{R}^2$ is given by $g((x,y),t)=\frac{dx^2+dy^2}{e^{4t}+x^2+y^2} \;\;\; (\star) $ Here are my ...
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404 views

Determining the “positivity” or “negativity” of Chern class (number?) of zero-sets of homogeneous polynomials

If $\Omega$ is the curvature 2-form on a $n-$manifold, then I would think that the Chern classes (forms), $c_k$ are defined as, $det(I + \frac{it\Omega}{2\pi}) = \sum c_k t^k$ I would like to ...
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1answer
287 views

question on connection on tensor bundle induced by a linear connection

I have a question regarding the definition of the covariant derivative of tensor fields, as given by John Lee in the book Riemannian Manifolds: An Introduction to Curvature On page 53 he states the ...
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167 views

Riemannian connections: how to understand $\overline{\triangledown}_X \langle Y,Z \rangle$

I have a question regarding a comment in Lee's book "Riemannian Geometry - an Introduction to Curvature". On page 52, Lee introduces the Euclidean Connection as the map $\overline{\triangledown} : ...
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1answer
260 views

Upper bound on covering multiplicity

Suppose $M^n$ is a manifold with $Ric(M)\geq(n-1)k$ for some $k \in \mathbb{R}$. For a given point $p \in M$, let $B_p(r)$ denote the metric ball of radius $r$. Given $B_p(r)$ and $\varepsilon > ...
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392 views

How to calculate scalar curvature in a local chart

I want to find a complete manifold with infinite diameter which has uniformly positive scalar curvature. And I want to show that $M^n = S^2(r) \times \mathbb{R}^{n-2}$ with $n \geq 3$ is an example ...
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201 views

Riemannian Connection (Very basic question)

We know that a connection $\nabla$ in a manifold M hashas the purpose of performing the same role as the covariant derivative of vector fields of surfaces in $\mathbb{R}^3$. Such analogies are ...
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2answers
148 views

help on connections

In the book on Riemannian Geometry by John Lee ("Riemannian manifolds: an Introduction to Curvature") the author gives an exercise on page 54 involving connections: Let $\triangledown$ be a linear ...
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1answer
391 views

Need references on Cartan's method of moving frames.

Could anyone suggest a book or a paper containing a good, modern treatment to the Cartan's method of moving frames. Especially, I am interested in its use in studying geometric properties of surfaces ...
2
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1answer
315 views

The behavior of all unit speed geodesics on a surface of revolution.

In the $xz$-plane of $\mathbb R^3$, consider the closed non-singular curve $\gamma$ which is the image of the function $$t\mapsto (1+2\sin^2(t))(\cos(t),0,\sin(t)).$$ (Note that $\gamma$ is invariant ...
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227 views

Proof of Gauss's lemma in Riemannian geometry

In the proof of Gauss's lemma here, there is a step $\displaystyle\lim_{t\to0}\frac{\partial f}{\partial s}(0,t)=\lim_{t\to0}T_{tv}\ \exp_p(tw_N)=0$ However, the limit seems meaningless (unless ...
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2answers
325 views

Notation for covariant derivative

I'm reading John M. Lee's book " Riemannian Manifolds". On page 57, the covariant derivative of $V$ along a curve $\gamma$ is defined, where $V$ is a vector field along $\gamma$. It is denoted by ...
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1answer
171 views

Flat metric is unique up to diffeomorphism

With respect to answer, I am not able to see the following: " Flat Metric Is Unique Up To Diffeomorpshim " What i meant can be seen by clicking the link. Is it trivial? Can someone help me out.
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2answers
197 views

Isometry and Immersion between two riemannian manifolds

I am confuse with these concepts: Isometry and Immersion. Let $M$ and $N$ be riemannian manifolds. If $f:M\to N$ is a smooth isometry and will it be a immersion... If $g$ is a immersion then i know ...
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1answer
70 views

Proving that inversions are isometries with respect to the hyperbolic metric.

I'd like to prove that the standard inversion $$(r,\theta)\mapsto\left(\frac{1}{r},\theta\right)$$ is an isometry with respect to the hyperbolic metric on the upper half-plane, and it would be nice to ...
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1answer
418 views

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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1answer
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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1answer
421 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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1answer
368 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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198 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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439 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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1answer
77 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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1answer
360 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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1answer
131 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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1answer
412 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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2answers
433 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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120 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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1answer
84 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
93 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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118 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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0answers
137 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 ...
12
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2answers
449 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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1answer
283 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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1answer
1k 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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112 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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0answers
262 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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3answers
414 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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193 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 ...