(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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Curve on Riemannian Manifold

A curve on Riemannian Manifold is $c:I\rightarrow M$. We study many properties about it, like parallel $\bigtriangledown_\dot{c}X=0$ and geodesic $\bigtriangledown_\dot{c}\dot{c}=0$. And we apply the ...
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2answers
97 views

On a Riemannian manifold (along a geodesic), is the relation *is conjugate to* transitive?

Let $(M, g)$ be a complete Riemannian manifold. Suppose $\gamma : \mathbb{R} \rightarrow M$ is a geodesic such that the instant $0$ is conjugate to both $a$ and $b$, where the numbers $a, b, 0$ are ...
3
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2answers
77 views

What is the value of the 2-form $X(u(Y))-Y(u(X))-u([X,Y])$, is it $2du(X,Y)$?

Let $X,Y$ be two vector fields on a Riemannian manifold $(M,g,\nabla)$ where $\nabla $ is the symmetric metric connection on $M$ and let $u$ be a 1-form. I want to find the value or a simple form or ...
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the Definition of Connection

Let $M$ be an Riemannian Manifold and $\bigtriangledown$ be the Riemannian Connection on it. Let we think about the domain and range of $\bigtriangledown:\Gamma(M)\times\Gamma(M)\rightarrow\Gamma(M)$ ...
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54 views

Example of '$g$' which is not a metric on $S^2$.

I am trying to find out a non-degenerate, positive, bilinear form defined for every point $p$ in $S^2$, such that it is not a metric and illustrate the same (i.e. it must not be satisfying the ...
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1answer
49 views

Ruled surface defined by a Exponential map

I have a regular curve $\alpha(s)$ on a 3-D Riemannian Manifold, with Frenet frame $\{T,N,B \}$. I define a surface now as $$X_N(s,t) = exp_{\alpha(s)}(tN(\alpha(s)))$$ $exp_p$ being the ...
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1answer
109 views

Show that the Lie derivative is equal to the commutator

Let $\Omega \subseteq \mathbb{R}^d $ be open. Let $\epsilon > 0$. Let $(\phi_t)_{t \in (-\epsilon , \epsilon)} $ be a family in $\mathrm{Diff}(\Omega)$ such that $ \phi_0 = id_{\Omega}$ and ...
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1answer
100 views

What is the importance of Metric in Riemannian Geometry?

Any smooth manifold $M$ is locally diffeomorphic to an open set in $\mathbb{R}^N$. So the tangent space at each point $p \in M$ is also isomorphic to $R^N$ where $p$ is mapped into the origin. So we ...
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1answer
64 views

Confusing definition of Jacobi field

Let $\mathcal{M}$ be $n$-dimensional Riemannian manifold. In wikipedia article I've found that a vector field $J$ along a geodesic $\gamma$ is said to be a Jacobi field if it satisfies the Jacobi ...
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1answer
61 views

Two linear conections on a Riemannian manifold and one … !!!

Let $(M,g)$ be a Riemannian manifold and $\nabla$ and ${\nabla}^*$ be two linear connections. If $f$ is a real valued function defined on $M$ and $X$ is a vector field, my question is: Does the value ...
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239 views

Resources for properly developing a modern understanding tensors

I am currently learning about tensors as they come up in the mathematics behind continuum mechanics. I was fairly disappointed with my initial foray into tensors, as presented in the book Classical ...
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118 views

The Riemannian Metric On Sphere.

I do a homework: Riemannian metric in the projective space. I feel confused: when we mention the Riemannian Metric, what are we talking about? If someone give you a sphere, it is just a set and ...
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1answer
28 views

Is $C^{\infty}(M) \subseteq L^2(M, \text{loc})$?

Let $M$ be a Riemannian manifold. Is it true that every smooth function on $M$ is also in $L^2(M, \text{loc})$? If so, could you give me some hint as to how to prove it or suggest a reference where I ...
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0answers
47 views

Metric on an open subset of $\mathbb{R}^d$ and Christoffel symbol of the second kind under diffeomorphisms

This is a follow-up question to this question I proved the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the ...
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1answer
34 views

The gradient estimate of the partition of unity

If $M$ is a compact Riemannian manifold with metric $g$, can we find a constant $C>0$, which is independent of $M$ and $g$, such that for any finite open covering $\{U_i\}$ of $M$, we can find a ...
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78 views

Intuition about the Riemann and Ricci tensors

I have started an attempt to self-study Riemannian Geometry. I well understand all the algebraic properties of the Riemann tensor (With a symmetric connection), and how it gradually becomes less and ...
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3answers
450 views

What is the definition of $R_{ijkl}$ in terms of metrics on a manifold?

What is the definition of $R_{ijkl}$ in terms of metrics on a manifold? I know what the definition of the riemann tensor, $R^l_{ink}$, is. But what exactly is meant by $R_{ijlk}$?
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1answer
138 views

The Definition of the Second Fundamental Form

Let $r:M\rightarrow{\mathbb{R}^{n+1}}$ be an isometric immersion and $M$ is an $n$-dimensional Riemannian Manifold. That is to say, $M$ is the hypersurface in $\mathbb{{R}^{n+1}}$. Then we can ...
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0answers
55 views

Finding the components of the Riemannian tensor given the components of a metric.

I am looking at a manifold of dimension $n$ (And I am considering a local co-ordinates system $x_1,x_2,\ldots x_n$) and the metric defined by the components $g_{ij} = \frac{\delta_{ij}}{x_1^2}$. I'm ...
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2answers
100 views

Are these two 2-manifolds homeomorphic?

I have a 2-Sphere with a finite number $k$ of points removed (at least 3), and I want to equip it with a Riemannian metric of constant negative curvature. My first thought was to take a free ...
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1answer
56 views

Elliptic Regularity on Manifolds

Sorry if this has been asked before. Are there are books which talk about elliptic regularity on manifolds? Everything I've been able to find has been about $\mathbb{R}^n$. Can every question about ...
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1answer
287 views

Another vector bundles over of a Riemannian manifold

How is that any other vector bundle over a Riemannian manifold can be turn into a Riemannian manifold as well? I think that the question is of interest due to the presence in math.SE of several ...
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1answer
98 views

Submanifold is complete

If $M$ is a complete manifold and $N\subset M$ is a closed, embedded submanifold with the induced Riemannian metric, show that $N$ is complete. I really don't know where to start. This is not ...
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2answers
127 views

Riemannian geometry: …Why is it called 'Exponential' map?

The exponential map $exp_{p}:T_{p}M \to M$ given a suitable $v \in T_{p}M$, returns $p$, displaced along the geodesic uniquely determined by $(p,v) \in TM$ for unit "time". So, what does the above ...
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2answers
103 views

Does this orbifold embed into $\mathbb{R}^3$?

Let $X$ be the space obtained by gluing together two congruent equilateral triangles along corresponding edges. Note that $X$ has the structure of a Riemannian manifold except at the three cone ...
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122 views

Definition of a differentiable manifold and papers in Riemannian geometry

There are at least two ways of introducing a definition of differentiable manifolds. I read John Lee's excellent book "Introduction to smooth manifolds" before, but there is too much bundles there for ...
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1answer
56 views

Metric tensors. Have I got the correct understanding?

My course is covering metric tensors in a slap-dash way, so I want to ensure I have understood correctly how they are described. I hope you can confirm this! So, I believe that a metric tensor is a ...
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1answer
260 views

Riemann tensor in terms of the metric tensor

The Riemann tensor is a function of the metric tensor when the Levi-Civita connection is used. Most tensor notation based texts give the Riemann tensor in terms of the Christoffel symbols, which are ...
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1answer
82 views

A question linked with concept of Lie derivative

Suppose $M$ is a Riemannian 3-manifold. We introduce a function $t$ on $M$ such that the two dimensional surfaces "$t=\text{constant}$" in $M$ are nested topological 2-spheres with the innermost ...
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364 views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
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28 views

Computing parallel transport for Hyperbolic plane

I am just rephrasing an earlier question of mine. I would like to know how one goes about calculating the parallel transport for a simply connected Hyperbolic plane which has a sectional curvature of ...
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68 views

Expressions for exponential map and parallel transport

This came up in a paper I was perusing. The authors list three formulas which I have not been able to comprehend. Here M is supposed to be a simply connected space form. Then for the exponential map ...
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55 views

A question about the Euler characteristic

Let $G$ be a finite group acting freely on a compact and orientable Riemannian manifold of dimension 2. I want to show that $\chi(M/ G)=\frac{\chi(M)}{|G|}$, where $\chi$ is the Euler characteristic, ...
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1answer
179 views

Metric on an open subset of $\mathbb{R}^d$ and Christoffel symbol of the second kind

I need to prove the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the Christoffel symbol of second kind. $g^{ij}$ ...
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1answer
130 views

Covering a Riemannian manifold with geodesic balls without too much overlap

I'm looking for a proof of the following fact: Let $M$ be a compact Riemannian manifold. There is a natural number $h$, such that for any sufficiently small number $r>0$, there exists a cover of ...
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78 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
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0answers
48 views

The relation between conformally related metrics and conformal vector fields?

Two metrics $g_{1}$ and $g_{2}$ are conformally equivalent metrics if $g_{2}=e^{2\theta}g_{1}$ A vector field $X$ is called conformal if $L_{X}g=2\theta g$ where $L_{X}$ is the Lie derivative with ...
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What can we say about the integral curve of a vector field on the warped product manifold?

Let $Z=(X,Y)$ be a vector field defined on the warped product $M×_{f}N$ where $f$ is defined on $M$. The integral curve of $X$ on $M$ is $\alpha$ and the integral curve of $Y$ on $N$ is $\beta$. I ...
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1answer
87 views

Is every Lie group the automorphism group of a riemannian manifold?

Given a finite-dimensional Lie Group $G$, is there always a Riemannian manifold $M$, such that $G$ is the group of isometries of $M$?
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2answers
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Please explain the shortest path between two points in non-euclidean geometry. [closed]

Please explain it for those with inferior knowledge of mathematics (using easy to understand words): e.g., kids and adults with no knowledge of mathematics (calculus, algebra , etc.) , or rather from ...
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2answers
130 views

when can you estimate curvature from finite information about two geodesics?

Let $c_v, c_w$ be two geodesics starting at a point $p\in M$, where M is a nonpositively curved, complete, smooth Riemannian manifold. Say $c_v(\varepsilon) = \exp_p(\varepsilon v)$ and ...
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1answer
114 views

What is the importance of conformal vector fields on Riemannian manifolds?

A vector $X$ on a Riemannian manifold $(M,g)$ is called conformal if $L_{X}(g)=2sg$ where $L_{x}$ is the Lie derivative and $s$ is a real-valued function on $M$. If $s=0$, $X$ is called a killing ...
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1answer
43 views

Euler characteristic of 2-dimensional compact Lie Groups

I'd like to know why the Euler characteristic of $G$, a compact Lie Group of dimension 2, is zero. I'm aware of the fact that this is true not only for dimension 2. The point is that I'm not familiar ...
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1answer
80 views

How many degrees of freedom does a metric have on a psuedo-Riemannian manifold?

I know this not that well posed of a question so please bear with me. Suppose we have a $n$-dimensional psuedo-Riemannian manifold $(M,g)$. We have that there are $n^2$ functions that make up ...
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1answer
30 views

Validity of a simple equation

Suppose $(\Sigma,h_{ij})$ is a 3 dimensional Riemannian manifold and $S$ is a 2 dimensional submanifold of $\Sigma$. Is the following equation true? $$2(\nabla_i R^{ij})n_j=(\nabla_kR)n^k$$ where ...
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2answers
75 views

Why is this curve a topological manifold?

Why is $$M=\{(z_1,z_2)\in \mathbb{C}^2 \, |\,\, z_1^3-z_2^4=0 \}$$ a topological manifold? I understand for example why why $|z|=1$ is a topological manifold, since I can write every point as ...
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1answer
82 views

Why are affine connections called so?

I have been going through some books regarding Smooth Manifolds and Riemannian Manifolds. But I haven't been able to get an answer to one question. Could you explain what is the intuition behind ...
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1answer
50 views

generalized warped product

Let $(M_1,g_1)$, $(M_2,g_2)$ denote two Riemannian manifolds, let $(I,dt^2)$ be the unit interval with its standard metric. I would like to study the manifold $(M,g)$ where $M = I \times (M_1 \times ...
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1answer
128 views

Intersecting geodesics in a positive curvature manifold

Suppose $M$ is a connected, compact orientable 2-dimensional Riemannian manifold, with positive Gaussian curvature. I'd like to show that two non-self-intersecting closed geodesics must intersect each ...
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Triangle Mesh as a topological disk [duplicate]

I was reading up on the Dirichlet problem, and was truly hoping if anyone here has the time to help make me understand this a bit better. In particular, the question relates to harmonic maps. My ...