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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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) ...
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58 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 ...
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26 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 ...
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39 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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59 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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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?
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71 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 ...
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46 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 ...
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37 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, ...
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98 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 ...
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49 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 ...
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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 ...
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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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15 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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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} ...
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51 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 ...
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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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23 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 $(M, g)$ ...
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42 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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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 ...
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2answers
59 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 ...
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47 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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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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28 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 ...
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41 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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93 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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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
53 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 ...
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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 ...
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28 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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13 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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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 ...
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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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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 ...
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36 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 ...
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28 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 ...
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36 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, ...
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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 ...
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25 views

A constant curvature manifold is Einstein

I would like to prove that a Riemannian manifold $(M,g)$ with constant curvature $k$ is Einstein, with Einstein constant $(n-1)k$. My attempt: Let $e_1, \dots e_n$ be an orthonormal basis for $T_pM$, ...
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Existence of a Lorentzian manifold

Does there exist a smooth compact Lorentzian manifold $M^n$ with n > 2, that has constant curvature, is simply connected and has Euler number (Euler characteristic) $\chi(M)=0$?
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15 views

Hopf-manifolds are complete

Let $M=\mathfrak{R}^2-{0}$ be a manifold equipped with the metric \begin{equation} g=\frac{\langle,\rangle}{x^2+y^2}, \end{equation} where $\langle,\rangle$ is the standard Euclidean metric. Let ...
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10 views

Schur Lemma (Riemannian Geometry)

Let $(M, g)$ be a Riemannian manifold. Assume that $$ \text{sec}_p(\pi) = f(p) \qquad\text{ for all } 2-\text{planes } \pi \subset T_pM \text{ and for all } p \in M. $$ Where sec$_p(\pi)$ is the ...
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Extension to a normal field

Let $M$ be a submanifold of the riemannian manifold $\overline{M}$, with the induced metric. Denote by $\nabla$ and $\overline{\nabla}$ the riemannian connections of $M$ and $\overline{M}$, ...
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44 views

Lenght of the curve in Riemannian metric.

Let $M^{k}$ a submanifold, $h:U\to M^{k}$ a chart, and $\gamma:[a,b]\to h(U)\subset M^{k}$ a curve in $M^{k}$. Represent the curve in coordinates $(h,U)$ as ...
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65 views

Visualizing the evolution of a Riemannian metric

I'm doing some reading into Riemannian geometry and PDEs and I have the following question. When we evolve a Riemannian metric (by say the Ricci flow) we are evolving a bilinear form on a manifold ...
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225 views
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Exponential map on the ellipsoid.

Consider the ellipsoid $M \subseteq \mathbb{R}^3$ defined by $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{x^2}{c^2} = 1,$$ where $0 < a < b < c$, equipped with the usual Riemannian metric ...
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47 views

Curvature and topology

I am studying Riemannian Geometry and I came across various Theorems which give conditions on the topology of a manifold given conditions on curvature, and vice-versa. Just to mention a few of them: ...
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37 views

Can a conformal map be turned into an isometry?

Let $f: (M, g) \to (M, g)$ be a conformal diffeomorphism of the riemannian manifold $(M, g)$, with $$ g(f(p))(Df(p) \cdot v_1, Df(p) \cdot v_2) = \mu^2(p) g(p)(v_1, v_2), \quad \forall p \in M, \, ...
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21 views

Asymptotic behaviour of the riemannian metric in polar coordinates

I'm studying the section 7 ("Local Geometry in Constant Curvature) of chapter 5 of "Riemannian Geometry" written by Petersen. At the beginning there is a Lemma which says how behaves the metric $g$ ...
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2answers
57 views

Invariance of function under isometry of Riemannian manifolds

Suppose that $(M,g)$ and $(N,g')$ are Riemannian manifolds and that $f: M \to N$ is an isometry. Now take smooth vector fields $X, Y, Z$ on $M$. Is it true that $X\langle Y, Z\rangle_p = ...