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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Algebraic topology & Riemannian geometry project idea?

I'm taking a first course on Riemannian geometry this semester. For a final project, I would like to do something that involves algebraic topology. However, the only results I know in algebraic ...
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0answers
34 views

Proof of the Beltrami theorem

I'm studying from a textbook and came across a theorem as the following, which it calls the Beltrami Theorem: Beltrami Theorem: A Riemannian metric $g$ is projectively flat if and only if it is ...
3
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1answer
41 views

Laplacian of a distance function on a Riemann manifold

For some reasons I need to show the following fact. Let $(M, g)$ be a Riemannian manifold. Let $U \subset M$ be an open set and $r: M \to \mathbb{R}$ a smooth distance function. Let us assume ...
2
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51 views

Computing connections on manifolds

Let $\nabla$ denote the Levi-Civita connection on the following manifold in $\mathbb{R}^3$ with Riemannian metric $g$ as follows: \begin{equation} \mathcal{H}^3=\lbrace (x,y,z)\in \mathbb{R}^3 \mid ...
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45 views

Motivation for Non-Euclidean geometry: relativity

I'm looking for references to motivate the study of non-Euclidean geometry. In particular I would like something about relativity. I do not want texts to learn non-Euclidean geometry, only ...
3
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1answer
31 views

Extending parallel 1 forms to harmonic forms on a compact set

Based on this question from Peterson's Riemannian Geometry: Let $(M,g)$ be an n-dimensional connected Riemannian manifold that is isometric to Euclidean space outside some compact subset $K ...
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1answer
43 views

Proving that Levi-Civita connection is preserved by isometries

I am trying to prove that given two Riemannian submanifolds $S,S'$ with Levi-Civita connections $\nabla , \nabla'$ and an isometry $f$, then $$ Df(\nabla_XY)=\nabla'_{X'}Y' $$ where, ...
-1
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1answer
62 views

Orthonormality of vector fields

I want to show that the following Riemannian manifold with given metric $g$ the following vector fields are orthonormal at every point $p$ of the manifold. Let $\mathcal{H}^3=\lbrace (x,y,z) \in ...
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1answer
33 views

Computing the Lie brackets of vector fields on a specific Riemannian manifold

Let $\mathcal{H}^3=\lbrace (x,y,z) \in \mathbb{R}^3 \mid z>0\rbrace$ be equipped with the Riemannian metric: \begin{equation*} g=\frac{dx^2+dy^2+dz^2}{z^2} \end{equation*} And consider the vector ...
2
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2answers
45 views

Levi-Civita Connection and vanishing christoffel symbols

Is there a good way to guess for what indices christoffel symbols, $\Gamma_{ij}^k$ vanish in general? For example, when calculating the Levi-Civita with spherical coordinates for a sphere most ...
0
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0answers
36 views

Dual basis cotangent space

I have been given the unitary sphere in the Euclidean space. $$F(\theta, \phi) =(\sin\theta \cos\phi, \sin\theta \sin\phi,\cos\theta)$$ I'm asked to show that the dual base of $E_1=F_*(\partial ...
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1answer
49 views

Reference about Karen Uhlenbeck

When I read Hamilton's 'FOUR-MANIFOLDS WITH POSITIVE CURVATURE OPERATOR',I am curious the details about Karen Uhlenbeck trick. But I can't find suitable reference to read it . What should I read ...
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1answer
42 views

Gaussian curvature versus sectional curvature

I was studying https://en.wikipedia.org/wiki/Gaussian_curvature (exact version https://en.wikipedia.org/w/index.php?title=Gaussian_curvature&oldid=709607678 ) and there it says: (bold added) ...
0
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1answer
60 views

A notion of nonpositive curvature for general metric spaces

The proof of the following result should be done by using the second variation formula of geodesics but I do not know how to start or what is the main idea of the proof. (Lemma 3.7 in the paper: A ...
1
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1answer
34 views

Counting independent components of Riemann curvature tensor

I'm having some trouble understanding the counting procedure for the number of independent components of Riemann curvature tensor $R_{iklm}$ in 4D spacetime. (The answer is supposed to be 20, but I'm ...
4
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1answer
41 views

Coordinates on a Riemannian manifold given by a distance function

I am currently studying the book "Riemannian Geometry" by Petersen. Defintion: Let $(M, g)$ be Riemannian manifold and let $U \subset M$ be an open set. A function $r : U \to \mathbb{R}$ is said ...
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1answer
66 views

Can we prove uniformization by solving the Yamabe problem directly?

One version of the uniformization theorem says that a simply connected complex manifold is biholomorphic to either the unit disc, $\Bbb C$, or $\Bbb{CP}^1$. The proof of this goes through potential ...
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0answers
18 views

Differential geometry with mathematica [closed]

Do you know some good packages for Mathematica for doing differential geometry calculations like tensor computations, curvature, ecc?
3
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0answers
80 views

Relation between the Hessian and Laplacian

Let $(M^{n},g)$ be a smooth Riemannian manifold with a smooth boundary boundary $\partial M$. Assume that the Ricci curvature of $M$ is $Ric^{M}\geq0$, and the second fund. form of $\partial M$ is ...
1
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1answer
47 views

Isometric spheres in euclidean space

I would like to prove that the sphere of radius $R>0$, $S^2(R)\subset \mathbb{R}^3$, with the induced metric is isometric to the sphere with radius $1$, $S^2\subset \mathbb{R}^3$, furnished with ...
4
votes
1answer
42 views

Riemannian metric given in polar coordinates

the Riemannian metric of the euclidean plane is given in polar coordinates as \begin{align*} ds^2=dr^2+r^2d\theta^2. \end{align*} Consider more generally, \begin{align*} ds^2=dr^2+\psi(r)^2d\theta^2, ...
5
votes
1answer
99 views

Intuition about the lack of a quadratic term in geometric expansions

Let $(\Sigma,g)$ be a Riemannian 2-manifold and let $p\in\Sigma$. It turns out that the circumference $C(r)$ of a geodesic circle $S_r(p)$ of radius $r$ around $p$ satisfies $$ C(r)=2\pi ...
0
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1answer
53 views

What is a Killing tensor?

Wikipedia gives the definition of a Killing tensor. Unfortunately, I don't know how to interpret the parentheses (it is also not explicitly explained in the link) and was therefore wondering whether ...
0
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0answers
20 views

Contracting products of bilinear form

In the picture below ,I don't know how to contract the $M_{\alpha\beta}$. I just used to the contract in Lee's book $$ tr:T^{k+1}_{l+1}\rightarrow T^k_l \\ F\rightarrow ...
0
votes
0answers
37 views

Invariance of a Vector Field under the action of a Group

I've got a one-parameter group given by \begin{equation} \theta_{t}\left(x,y,z\right)=\left(e^{t}x,e^{t}y,e^{t}z\right) \end{equation} I already have th infinitesimal generator vector field ...
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0answers
16 views

Under Ricci flow, how to show $\partial_t\int_{U_x} d(x_0,exp_xv) dv =0$?

Let $(M,g)$ be a Riemann manifold, and g evolving under $\partial_t g_{ij}=-2R_{ij}$. $U_x=\{v\in T_xM:v<r\}$ , $r$ is injectiion radius, $d(~,~)$ is distance function, $exp$ is exponential map. ...
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0answers
25 views

Let $ f(x)=\int_{U_x} d(x_0,exp_xv) dv $ ,how to show $\nabla f$ and $\nabla^2 f$ are bounded?

Let $(M,g)$ is a Riemann manifold, $x_0$ is a point of $M$. $U_x=\{v\in T_xM:v<r\}$ , $r$ is injectiion radius, $d(~,~)$ is distance function, $exp$ is exponential map. Set $$ f(x)=\int_{U_x} ...
3
votes
1answer
56 views

If $f: M \to N$ is a diffeomorphism and $N$ is complete, then $M$ is complete

This is a problem from Riemannian Geometry by Do Carmo, namely Ch. 7, Sec. 3, Problem 7 on pg. 153. Let $M, N$ be Riemannian manifolds with $N$ complete, and $f: M \to N$ a diffeomorphism for which ...
3
votes
1answer
39 views

Locally disk-preserving charts?

This is slightly vague as I've not yet come to terms with what I'm actually looking for. On $S^2$ we may choose charts (stereographic projection) such that the image of a disk (i.e. all points ...
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1answer
25 views

Conformal change of Riemannian metric

I'm studying Riemannian Geometry from different sources and I have a problem trying to solve one of the exercises from Petersen's Riemannian Geometry: Show, that any Riemannian Manifold $(M, g)$ ...
1
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2answers
48 views

Proof of the contracted Bianchi identity

In proving the contracted Bianchi identity, I have problems understanding the contractions. Starting with the second Bianchi identity: $$R_{ijkl;m}+R_{ijlm;k}+R_{ijmk;l}=0$$ The first step is to ...
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0answers
26 views

Smooth of $d(x_0,exp_x(v))$ and injectivity radius.

According to this question, the curvature can't control the injectivity radius. So, I don't know why support of $\varphi(v)$ need be small compared to maximum curvature? I think it should be compared ...
2
votes
2answers
78 views

Affine connection, metric and parallel transport and mutual interdependence

I am eternally confused even after repeated learning about the mutual independence between affine connections and the metric tensor and parallel transport. Given any one of them, can I recover the ...
3
votes
1answer
49 views

Gauss formula for a 3-sphere embedded in $\mathbb{R}^4$

Given connections $\nabla$ and $\bar{\nabla}$ as connections on $\mathbb{R}^4$ and the 3-sphere of radius $r$: $\mathbb{S}^3(r)$, the vector fields $X,Y$ tangent to $\mathbb{S^3}(r)$, how do I obtain ...
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0answers
22 views

Covariant differentiation of a vector field

Let $F=\alpha + \beta$ be the Randers metric on manifold $M$ of dimention $n$. Here $\alpha$ is a Riemannian metric on $M$ and $\beta$ is a 1-form on $M$. Let $V=V^i \frac{\partial}{\partial x^i}$ be ...
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1answer
30 views

Understanding “trace of map” in the definition of harmonic maps

I have difficulty understanding "trace of map" in the definition of harmonic map. Let $\phi: (M,g)\to (N,h)$ is map between two Riemannian manifolds, the energy density is defined as ...
2
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0answers
44 views

Calculation of extrinsic curvature

I asked this question first on physics.SE but I got no complete answer so I thought maybe someone here could help. I'm trying to understand how to derive the extrinsic curvature (in order to ...
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1answer
95 views

Strange coarea formula on the cylinder. Is it correct?

Consider the cylinder $[-1,1]\times S^{1}$, where $S^{1}=\mathbb{R} / \mathbb{Z}$ and let $x$ be the coordinate on $[-1,1]$ and $y$ the coordinate on $S^1$. Let $\alpha$ be a 1-form on $[-1,1]\times ...
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2answers
43 views

Proof of Wikipedia formula about Ricci curvature

In the Wikipedia article on Ricci curvature there is a formula, the third of the paragraph "Direct geometric meaning", that reads: $$ d\mu_g = \Big[ 1 - \frac{1}{6}R_{jk}x^jx^k+ O(|x|^3) \Big] ...
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1answer
60 views

Why does the Levi-Civita connection commute with pullbacks and pushforwards?

If $i: M \to N$ is an embedding of Riemannian manifolds, I am trying to prove that $\nabla i^* T = i^* \nabla T$ for any covariant tensor $T$ (I use the same letter for the two Levi-Civita ...
3
votes
2answers
58 views

Derivative along a curve

Suppose $M$ is a hypersurface of the sphere $S^n \subset \mathbb{R}^{n+1}$, and denote the riemannian connections of $M$, $S^n$ and $\mathbb{R}^{n+1}$ by $\nabla, \overline{\nabla}$ and ...
0
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0answers
30 views

A system of differential equations with no solution

According to book "Differential geometry of spray and Finsler spaces", the following spray on manifold $\mathbb{R}^2$ can not be induced by a Finsler metric: $$G = u \frac{\partial}{\partial x} + v ...
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14 views

Basis vectors for “perturbed slicings” of a function, using SE(3)

Given a function $\Phi: \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\mathbb{R}^2\rightarrow\mathbb{R}$, using a rigid mapping $s \in SE(3)$ such that for ...
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1answer
32 views

What is meaning of word “mean” here?

This is a part extracted from a textbook (book "Riemann-Finsler geometry" by Chern & Shen): . My question: Why do we say that the tensor $\mathcal{J}$ (mean Landsberg tensor) is a mean of the ...
3
votes
0answers
22 views

Christoffel Symbols in terms of the Log Function

Since the Riemmanian Log function expresses the Manifold structure in terms of $\mathbb{R}^d$ locally, then I was wondering: Can we express the Christoffel Symbols explicitly in terms of the Log ...
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0answers
23 views

Extension of divergence free vector field as a divergence free vector field.

Let $M$ be a compact smooth Riemannian manifold of dimension $n$. Assume that $M$ is isometrically embedded in $\mathbb{R}^m$ for some sufficiently large $m$ via the map $\iota$. Let $X:M\to TM$ be ...
2
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0answers
43 views

Christoffel symbols in polar coordinates calculation

I'm currently studying Riemannian Geometry and I would like to get familiar with the basic concepts. I considered the simple Riemannian manifold $(\mathbb{R}^2, can)$ with its Levi-Civita connection ...
0
votes
1answer
29 views

Integration over one point set

Suppose we have a continuous function $f:S^2\rightarrow \mathbb{R}$ on the standard sphere. What value is given for the following integral $$\int_{\lbrace p\rbrace }f(x) \text{d$vol_{\lbrace ...
3
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0answers
43 views

Pulling back a Kähler structure on a symplectic submanifold

Let $(K, G, \Omega, J)$ be a Kähler manifold and $(S, \omega)$ be a symplectic manifold. Let $i : S \to K$ be a symplectic embedding. Is it possible to endow $S$ with a Kähler manifold structure, ...
4
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1answer
45 views

Riemmanian Distance is always greater?

Setup: Suppose $M$ is a $C^k$-manifold embedded into some Hilbert space $H$ and $g$ is the induced Riemmanian metric thereon (induced by restricting the inner-product $\langle,\rangle_H $ in $H$ to ...