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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441 views

Is the metric on the circle, induced from the plane, not a flat one?

My question concerns the highlighted part posted below, from Wikipedia article. (Link to the revision at the time of this post.) I'd say I can't detect the curvature of the unit circle if I go along ...
0
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1answer
56 views

How can we show that parallel transport is invertible?

We have that the map $\Pi^{pq}_{\gamma}: T_pS \rightarrow T_qS$ that takes $v_0 \in T_pS$ to $v_1 \in T_qS$ is called parallel transport from $p$ to $q$ along $\gamma$. $\Pi^{pq}_{\gamma}: T_pS ...
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0answers
28 views

Example for conjugate points with only one connecting geodesic

$\newcommand{\ga}{\gamma}$ $\newcommand{\al}{\alpha}$ I would like to find an example for a Riemannian manifold, that has two conjugate points $p,q$ with only one connecting geodesic between them. ...
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0answers
9 views

Birkhoff theorem and maximally symmetric submanifolds

I'm trying to understand the proof of Birkhoff theorem (in general relativity, on the spherically symmetric solutions of the vacuum Einstein equation). But I don't understand the beginning of the ...
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0answers
34 views

Manifolds, where its enough to have one chart for integration

Assume a compact connected manifold $M$ is given as a subset of some $\mathbb{R}^m$. Assume we have a chart $\gamma:U \rightarrow M$ such that $M-f(U)$ (the set $M$ without $f(U)$) has zero measure in ...
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2answers
67 views

How calculate with Riemannian metrics (e.g. Multiplication and Divison)?

I have no idea how to handle the following Riemannian metrics, how to find the estimates for the bound and how to actually calculate with $g$ and $d$. Do I need to use the matrix representation? Or ...
0
votes
0answers
23 views

Stereographic projection $S^3 \to \mathbb{P}^2(\mathbb{C})$

I think I can find a stereographic projection $S^2\setminus\{(0,0,1)\} \to \mathbb{P}^1(\mathbb{C})\setminus\{[0,1]\}$ using spherical coordinates: it should be something like this $$(\theta,\phi)\to ...
2
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0answers
31 views

Christoffel symbols vanishing in normal coordinates

Let $(M,g)$ be a Riemannian manifold, and let $(\varphi,U)$ be normal coordinates in $p\in M$. For every $v\in T_p M$, denote $\gamma_v :I_v \to M$ the maximal geodesic with initial point $p$ and ...
1
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1answer
41 views

Bounds for Riemannian manifolds

I am familiarizing myself with a new topic and want to calculate an actual example (no homework). So far, I have defined the parameter coordinates, the diffeomorphism and the pullback metrics, but I ...
2
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0answers
49 views

Limit of Riemannian manifolds is not Riemannian

I want to prove that $D$, standard unit ball in ${\bf R}^2$ with $|\ |$, with a metric $\| \ \|_1$ is a limit of Riemannian manifolds $X_i$. Here problem is to find $X_i$ (If necessary, all metrics ...
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0answers
30 views
2
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2answers
80 views

Metric tensor second derivative in terms of riemann curvature tensor NOT in a local coordinate system

I know that the Riemann curvature tensor, in a general coordinate system is: $$ R_{abcd}=\frac{1}{2}(g_{ad,bc}+g_{bc,ad}-g_{ac,bd}-g_{bd,ac})+(...) $$ The dots mean lower derivatives of $g$. I am ...
0
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1answer
29 views

Non-trivial vector field with divergence $0$ on $\mathbb{S}^{2}$.

I am looking for a non-trivial vector field $X$ on the sphere $\mathbb{S}^{2}\subseteq\mathbb{R}^{3}$ with the induced metric such that its divergence is $0$. I recall here the expresion of the ...
1
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0answers
35 views

Proof of Synge theorem without homotopy

Can the following statement be proven without the use of homotopy? If $(M,g)$ is a closed, orientable Riemannian manifold with even dimension $m$ and positive sectional curvature then $M$ is simply ...
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0answers
32 views

Conformality of stereographic projection

I have to solve the following exercise: Let $S^m(r)=\{x\in \mathbb{R}^{m+1}|\sum_i (x^i)^2=r^2, r>0\}$ and let $\Phi$ be the Stereographic projection $$\Phi: ...
0
votes
1answer
49 views

Triangulation of Torus in three different ways, but two of them are wrong.

In my geometry notes the writer states that the following two, are not triangulations for the torus: On the contrary this is a good triangulation: I tried to wrap a piece of paper in order the ...
0
votes
0answers
5 views

Variation of a d'alambertian operator

Let $M$ be a pseudo Riemannian manyfold, $H$ be function of a scalar curvature $R$. Assume that variation of the metric tensor and it's first derivatives is zero on the border $\partial M$. Which ...
2
votes
1answer
39 views

Distances in geodesic triangles

Let $a, b \in \mathbb{R}^2$ be two points in the plane and let $\Pi$ be their perpendicular bisector (see left figure). Let $c \in \Pi$ be any point and consider the triangle $\triangle abc$. Suppose ...
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0answers
22 views

the distribution of Martinet

The distribution of Martinet is $\Delta=ker\omega$ with $\omega=\Bbb dt-\frac{1}{2}y^{2}\Bbb dx$. let $X=\partial x+\frac{1}{2}y^{2}\partial t$ and $Y=\partial y$ The curvature $2$-forme is: ...
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0answers
27 views

Prove that $X = \sin \phi \,\partial_\theta + \cot \theta \cos \phi\, \partial_\phi$ is a Killing vector field

I'm trying to prove that $X$ (given in the title of the question) is a Killing vector field for a metric which is spherically symmetric and static. This is what I've done so far: The vector field ...
0
votes
0answers
25 views

Covariant derivatives of the Liouville vector field

I am frustrated, help me please. On page 46 of the book "An introduction to Riemann-Finsler geometry" by Bao&Chern&Shen, we have: ** Question :** Show that $$\nabla ...
2
votes
0answers
31 views

$H^1_0(M)$ for non-compact $M$

Consider a complete non-compact Riemannian manifold $M$. My question is, is it possible for a non-zero constant function $c$ to be in $H^1_0(M)$? My guess is, this should be possible when $M$ has ...
0
votes
1answer
43 views

Is $0 = \nabla_Xg(Y, -)(Z) = \nabla_X(g(Y, Z)) - g(Y, \nabla_XZ)$ if $\nabla g= 0$?

Suppose a manifold $M$ with a metric $g_{\mu\nu}$. I know that the covariant derivative (I'm assuming the connection induced by the metric) of a covector is given by: $(\nabla_X \eta)(Y)=\nabla_X ...
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0answers
48 views

Functions determine geometry … Riemannian / metric geometry?

Given a compact Riemannian manifold $(M,g)$, is there a subring of $C^{\infty}(M)$ that determines the isomorphism class of $(M,g)$? (In the same way that $C^{\infty}(M)$ determines the diffeomorphism ...
1
vote
1answer
39 views

Equivalence of definitions of Killing vector field

I read in wikipedia the following two definitions of Killing Vector $X$: $$\nabla_{\mu}X_{\nu}+\nabla_{\nu}X_{\mu}=0$$ $$ g(\nabla_Y X,Z)+g(Y,\nabla_Z X)=0$$ I have problems deducing the ...
2
votes
1answer
40 views

Sectional curvature in a paraboloid is always positive.

I'm working on Lee's book ''Riemaniann Manifolds an Introduction to Curvature''. One exercise (11.1) is about to see that the paraboloid given by the equation $y=x_1^2+...+x_n^2$ has positive ...
2
votes
1answer
42 views

Smooth self-map, criterion for preserving connected components, isometry, rotation matrix preserves.

Consider $\textbf{R} \times \textbf{R}^\times = \textbf{C} - \textbf{R}$ wit h the metric tensor $ds^2 = (dx^{\otimes 2} + dy^{\otimes 2})/y^2$. For any$$g = \begin{pmatrix} a & b \\ c & d ...
0
votes
0answers
35 views

The tensor product of two riemannian metrics

let $g=(g_{ij})$ and $h=(h_{ij})$ be two Riemannian metric on $\mathbb{R^{n}}$. consider the Riemannian metric $g\otimes h$ as a metric on $\mathbb{R}^{n^2}$. how can we describe the geometry of ...
0
votes
1answer
29 views

Covariant derivative of a constant vector field

I am trying to figure out that if $X$ is locally constant, i.e. in some coordinate system, the coefficient function is a constant function, then whether the covariant derivative of $X$ is 0. In ...
9
votes
1answer
80 views

What's the name of the surface and Is it a $C^2$ smooth surface? [duplicate]

what's the name of the surface? Is it a $C^2$ smooth surface? Its implicit equation is: $(x−2)^2(x+2)^2+(y−2)^2(y+2)^2+(z−2)^2(z+2)^2+3(x^2y^2+x^2z^2+y^2z^2)+6xyz−10(x^2+y^2+z^2)+22=0$
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votes
0answers
17 views

Curvature of a product of Riemannian manifolds

If $\mathcal{M}$ is a Riemannian manifold of constant curvature, is the manifold $\mathcal{M}^n$ with the product metric, of constant curvature? (and why?) Thank you
1
vote
1answer
40 views

How to show $d(x_0,\exp_x(v))$ is smooth?

$(M,g)$ is a smooth Riemann manifold.$d(\cdot,\cdot)$ is distance function on $(M,g)$. $x_0$ is a given point in $M$. $\exp$ is exponent map. If $v\in T_xM$ is enough small, how to show ...
2
votes
0answers
98 views

A calculate of curvature connecting intuitional images

When I read this Wiki (picture below is from it), it's so great that explain what the Riemann curvature tensor is.But I get stuck in the calculate in the red box. How to calculate it ? Because ...
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0answers
46 views

How to show $\nabla s(x)$ and $\nabla^2 s(x)$ are bounded? [closed]

Let $(M,g)$ is smooth complete Riemann manifold with bounded curvature. $exp$ is exponent map.For some given point $x_0\in M$ ,let $$ s(x)=d(x_0,exp_x(v)) $$ $d$ is distance function,$v\in T_xM$, and ...
2
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0answers
43 views

A specific example about computing of curvature.

After learned some Riemann Geometry, I want to compute the curvature of $S^2$ by the way of Riemann curvature. So, I assume the $S^2$ is $$ (\theta_1,\theta_2)\rightarrow ...
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0answers
22 views

What is the $N_{\alpha\beta}$?

Whether the $N_{\alpha\beta}=tr_{13}tr_{24}M_{ab}\otimes M_{cd}=M_{\alpha\beta}\otimes M_{\alpha\beta}$ ? Besides,in subsequent proof,$N_{\alpha\beta}$ satisfies $$ N_{\alpha\beta}v^\alpha v^\beta ...
0
votes
2answers
30 views

Metric of vector bundle of Riemann manifold.

Whether there is a standard metric of vector bundle of Riemann manifold? How is it defined ? Using the exponent map ? Or just using the metric of $\mathbb R^n$ ?
2
votes
0answers
36 views

Hessian comparison

I get stuck in the red line in picture below. First, although $\varphi$ is smooth, but seemly the derivertive about $x$ can't be moved to $\varphi$.So, I don't know why it's smooth. Second,I want to ...
0
votes
1answer
43 views

Volume form as eigenform of the Lie derivative

Suppose (M,g) is a homogeneous Lorentzian manifold and $Y$ a vector field on it. $G$ is a transitively acting Lie group. It is stated that the volume form $\omega$ is $G$-invariant. It is also stated ...
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vote
1answer
20 views

What is the easiest way to show that matric tensor of a Riemannian metric is positive-definite

Consider the following Riemannian metric: $$g_{ij}(x):= (1-\psi)\dfrac{(\delta_{ik}x^{k})(\delta_{jl}x^{l})}{|x|^{2}}+ \psi\delta_{ij},$$ where $$|x|:=\sqrt{\delta_{ij}x^{i}x^{j}}\qquad , \qquad ...
0
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0answers
34 views

Compute the curvature of $g = ds^2 + \frac1M \sinh^2(\sqrt M s) d\Omega^2$?

About 1.5 month ago, I ask this question . Today I think I can resolve it.Because I choice to think the easy case $n=3$.And in this case, I know what is the $d\Omega^2$, I'm confident. So , I do it . ...
1
vote
1answer
38 views

Liebmann-Süss Theorem

I have a question to the proof of the Liebmann-Süss theorem which is stated in my book as follows: Assume $x:M^n\rightarrow\mathbb{R}^{n+1}$ is a compact and connected oriented hypersurface, if $M$ ...
0
votes
1answer
34 views

Convex Manifold with constant mean curvature is hypersphere

I wish to understand the proof the following theorem: Assume $M$ is an unbounded, compact, connected, strictly convex $\mathbb{R}^{n+1}$ hypersurface. If the mean curvature of $M$ is constant, $M$ is ...
2
votes
2answers
81 views

Why does $\mathrm{tr}(\mathrm{ln}g_{\mu\nu})$ vary as $g^{\mu\nu}\cdot\delta g_{\mu\nu}$ under $\delta g_{\mu\nu}$?

For a pseudo-Riemannian manifold, under the variation $g_{\mu\nu}\mapsto g_{\mu\nu}+\delta g_{\mu\nu}$, the determinant $g=\mathrm{det}g_{\mu\nu}$ varies as $$\delta g=gg^{\mu\nu}\delta g_{\mu\nu}$$ ...
3
votes
1answer
60 views

Whether $df(X)=\langle\operatorname{grad}f,X\rangle$?

$(M,g)$ is Riemann manifold, $X$ is vector field, $f$ is function on $M$. $\langle\cdot,\cdot\rangle$ is inner. Whether $df(X)=\langle\operatorname{grad}f,X\rangle$ ? I only know $df(X)=X(f)$.
0
votes
1answer
34 views

Euler-Lagrange equation of energy of length function on Riemann manifold

$(M,g)$ is a Riemann manifold. $\gamma:[0,1]\rightarrow M$ is a curve.The length of $~\gamma $ is $$ L(\gamma)=\int^1_0 ||\dot\gamma (t)||_g ~dt $$ The energy is $$ E(\gamma)=\frac{1}{2}\int^1_0 ...
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votes
1answer
28 views

How to show $~\nabla _TT=0$?

$M$ is a Riemann manifold. $$ \begin{align} \alpha : & [a,b]\times(-\varepsilon,\varepsilon) \rightarrow M \\ &(t,s) \rightarrow \alpha(t,s) \end{align} $$ For any given $s\in [a,b]$, ...
0
votes
1answer
43 views

Variation under constraint

I always can't compute right.$u=u(x),R=R(x)$ and $\tau$ is constant, and $M$ is compact manifold.If $u$ is the minimizer of $$ \inf\{\int_M [\tau(4|\nabla u|^2+Ru^2)-u^2\ln ...
1
vote
1answer
20 views

Is every totally geodesic surface minimal?

Let $M$ be a Riemannian manifold, and let $S$ be a hypersurface (codimension $1$). If $S$ is completely geodesic, does that imply that it is minimal? If not, what are the conditions? If yes, is there ...
0
votes
0answers
14 views

Proof of an equality in Finsler Manifolds.

How can I to prove following equality in Chern-Bao-Shen (Riemann-finsler Geometry) Page 36? $$\ell_i\frac{\delta y^i}{F}=d(log(F))=\omega^{n+n}$$ and $$\frac{\delta F}{\delta x^i}=0\qquad\forall i$$ ...