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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Lie derivative on a riemannian manifold

Suppose we have a Riemannian manifold $(M,g,\nabla)$ with Levi-civita connection $\nabla$. We define a new symmetric non-metric connection $\bar\nabla$ on $M$. Then the Lie derivative of functions and ...
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38 views

Any books on isospectral manifolds?

I was searching stuff related to M.Kac's famous question "Can one hear the shape of the drum ?" I further found results due to Gordon, Webb and Wolpert in the 2D case using Sunada method. Are there ...
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401 views

What does it mean “being geodesic” is not invariant?

We know that "being geodesic" is not invariant under re-parametrization. Only affine re-parametrization preserves the property of being a geodesic. Also, a geodesic is locally distance minimizer. ...
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199 views

A new symmetric non-metric connection that generalizes the geodesic equation(Version 2)

A curve $\alpha$ on a riemannian manifold $(M,g,\nabla)$ is a geodesic if $\nabla_TT=0$, where $T$ is the tangent vector field. A generalization of this geodesic equation suggests that $\nabla_TT=\rho ...
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Wedge product of a one- form and a Kähler form

Let $x$, $y$, $v$, $w$ be coordinates on $R^{4}$ and $g$ be the Riemannian metric whose matrix with respect to these coordinates is $$g=\left ( \begin{array} {cccc} 1 & 0 & -kx & 0\\ 0 ...
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No constant curvature metric on $S^2 \times S^1$

I was reading the introduction to Hamilton's paper "Three-manifolds with Positive Ricci Curvature." He states that $S^2 \times S^1$ admits no metric of constant sectional curvature, and therefore ...
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107 views

Relation between two Riemannain connections

Let $g$ be a Riemannian metric on $M$ and let $\tilde{g}=f^{2}g$ where $f$ is a smooth function that is never zero. let $\nabla$ and $\nabla'$ be the Riemannain connections of $g$ and $\tilde{g}$ on ...
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101 views

What is Rotational on a Riemannian Manifold?

I have learned divergence, gradient and rotational in vector analysis of $\mathbb R^3$. However, when I read Riemannian Geometry, there are only definitions about divergence and gradient. So I have an ...
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87 views

Symmetry of the Riemannian curvature tensor

The Riemannian curvature tensor, in local coordinates, $R_{ijkl}$, has the following symmetries: $$R_{ijkl}+R_{jikl}=0;$$ $$R_{ijkl}+R_{ijlk}=0;$$ $$R_{ijkl}=R_{klij};$$ ...
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717 views

Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. ...
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318 views

Conformal Equivalence of two Riemann metrics

I'm reading a paper and encountered a concept of conformal equivalence between two Riemannian metrics on a differentiable $2$-manifold $M$ : Two Riemannian metric $g$ and $f$ are conformally ...
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148 views

Geodesics and manifold

I apologize if my post is "silly" because I don't know much about riemannian geometry. I know that $M = (0,1)$ (the open unit interval) can be seen as a one-dimensional manifold. Since $M$ is an ...
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129 views

Differential Geometry Notation

When we have a metric on a manifold, there is a natural isomorphism between the tangent space and the cotangent space, and so, if I understand correctly, it is not so important to keep track of which ...
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45 views

Reference for a theorem in Riemann geometry

I am watching an online lecture by John Morgan on lektorium, and at about 33:00, he claimed a theorem which I have never heard before, that There is a formula in terms of Riemann curvature $R$ ...
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86 views

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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118 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 ...
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87 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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61 views

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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63 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
64 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
175 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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180 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
164 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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72 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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282 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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1answer
34 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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54 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
52 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 ...
3
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131 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
462 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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192 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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64 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 ...
3
votes
2answers
120 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
147 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
397 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
145 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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179 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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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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136 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
83 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
922 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
92 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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837 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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120 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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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
249 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. $g^{ij}$ ...
7
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1answer
184 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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148 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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96 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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31 views

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 ...