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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82 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
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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 $df(X)=X(f)$.
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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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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
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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 ...
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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 $$\frac{\delta F}{\delta x^i}=0\qquad\forall i$$ ...
2
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0answers
25 views

Computation of the Ricci tensor and scalar curvature using Cartan's formalism

I've computed the Ricci tensor and the scalar curvature of the following metric on $I\times S^2, I\subset \mathbb{R}$: $$g=(1-r^2)^{-1}dr \otimes dr + r^2 d\theta \otimes d\theta + r^2\sin^2\theta ...
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0answers
36 views

Christoffel symbols for the Poincaré ball model

The metric tensor $g_{ij}$ of the Poincaré ball model is $$ g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2} $$ where $\delta_{ij}$ is the Kronecker delta and $x^k$ are the ambient Cartesian ...
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0answers
22 views

Where is the $\mu$ from?

When I try to compute the last equation, I get stuck. According to the last second equation, I have Lagrangian $$ L(p,z,x)=[\tau(4p^2+Rz^2)-z^2\ln z^2-nz^2](4\pi\tau)^{-\frac{n}{2}} $$ Then, $$ ...
6
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0answers
33 views

Sobolev space and integration by parts on non-orientable manifolds

Let $M$ be a compact manifold without boundary which is not orientable. Do all the standard facts that apply to oriented manifolds and Sobolev spaces also apply here? Like Green's formula for example. ...
3
votes
0answers
47 views

Meaning of / intuition for contraction of tensors (in the Riemannian setting)

I'm currently taking a course in differential geometry, and we are, I'm guessing, finally going to start working with the Riemannian curvature tensor after having covered a lot of smooth manifold ...
0
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0answers
16 views

Levi-Civita Theorem

" Theorem: Let ($M$,$g$) a Riemann variety.Then there exists a unique linear connection without torssion such that $\nabla g=0$$\qquad$$\qquad$ $\qquad$ $\qquad$$\qquad$Define:$(\nabla_X ...
2
votes
1answer
35 views

Compute using the define of Lie derivative.

The Lie derivative is defined as picture below.$X,Y,Z$ is vector fields,and $g$ is Riemannian metric. I try to compute $$ \mathcal L_Xg(Y,Z)=X(g(Y,Z))-g(\mathcal L_XY,Z)-g(Y,\mathcal L_XZ) $$ ...
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0answers
34 views

Polynomial functions on a smooth manifold

If one views $\Bbb R ^{2n}$ as the cotangent bundle of $\Bbb R ^n$, with coordinates $(q_1, \dots, q_n, p_1, \dots, p_n)$, then in order to do classical Hamiltonian mechanics on it one considers ...
2
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1answer
26 views

Is the interval [0,1] geodesically complete?

I am having some confusion about geodesic completeness. Is the interval [0,1] geodesically complete? As metric space it is complete hence Hopf Rinow theorem implies it is geodesically complete but a ...
4
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2answers
86 views

How to prove $d\omega=(\nabla_\mu\omega)_\nu dx^\mu\wedge dx^\nu$ without using coordinates

This is exercise 7.8 b) of Nakahara's GTaP: Let $\omega\in\Omega^1(M)$ be a 1-form on a Riemannian manifold with Levi-Civita connection $\nabla$. Prove that $$ ...
4
votes
1answer
31 views

If $M$ is Riemannian, then $\kappa_f \oplus f^*TN \cong TM$, where $\kappa_f$ is built out of kernels of the $Df_x$?

A smooth map $f: M \to N$ between smooth manifolds is a submersion if each Jacobian$$Df_x: DM_x \to DN_{f(x)}$$is surjective. I know how to construct a vector bundle $\kappa_f$ built out of the ...
0
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1answer
54 views

Differential Geometry. On the Hypotheses which lie at the Bases of Geometry

In that famous paper http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.html, Riemann writes the below. I get lost at the part in bold. Can someone explain what he means with an ...
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0answers
28 views

Definition of dual connection in riemannian geometry

If D is a connection on a vector bundle E, we define the dual connection D* so that $$d(v^*,w)=(D^*v^*)(w)+v^*(Dw)$$ I understand why this seems the natural thing to do. Why is the following not ...
3
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2answers
104 views

Introductory book on differential geometry for engineering major

I am an engineering major and looking for a straightforward, easy to understand basic book on differential geometry to get started. At starting point, I am not looking for a comprehensive book (may be ...
1
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1answer
49 views

Why vector field can be the gradient of a function $f$?

Because I always treat the gradient of $f$ ($\nabla f$ ) as $\nabla_ifdx^i$.And the vector field should be $X=X^i\frac{\partial}{\partial x^i}$. So I am fuzzy with that why vector field can be the ...
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0answers
30 views

How to show the inf can be achieved by some nonnegative $u\in H^1(M)$?

When I read some about Perelman's $\mathcal W$ function, I get stuck with the red line in the picture below.Seemly, I should to read the 8.2 Existence of minimizers of Evans' PDE. But I am not sure , ...
1
vote
1answer
38 views

Guess about the integral curve is not closed

For given smooth vector field $X$ on Riemannian manifold,solution of $\dot c=X(c)$ is the flow line or integral curve. Then the point $\lim\limits_{t\rightarrow\infty} c_p(t)$ need not be contained ...
2
votes
1answer
42 views

Each point is contained in precisely one integral curve

As Corollary 2.2.1 in picture below, each point is contained in precisely one integral curve. But it is obvious there are different integral curve which contain the point. I understand 'precisely one' ...
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0answers
19 views

A $C^2$ surface with constant mean curvature

Let us consider a manifold $M$ with metric $g$ and a surface $N_t$ in $M$ with mean curvature $H$ and second fundamental form $A$. Then if we assume that $H$ is a bounded constant $H(t)$ and $A = ...
2
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0answers
25 views

Classifying left invariant metrics on the 3-dimensional heisenberg group

Recently I read that all left invariant metrics on the Heisenberg group are equivalent up to scaling,however no reference was given for this result. I've made some attempt to prove this myself. In ...
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0answers
28 views

Intuitive understanding of quantum ergodicity of eigenfunctions

I recently heard a talk on differential geometry where the speaker was using a result called quantum ergodicity of eigenfunctions. I am trying to see if I am getting the gist of the result correctly. ...
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1answer
63 views

Curvature tensor for a particular Hilbert manifold

My question involves an infinite dimensional Hilbert manifold with a Riemannian metric. My question is: What is the form of the curvature tensor for a infinite dimensional Hilbert manifold with ...
0
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0answers
9 views

Sobolev Imbedding Theorem on manifold with boundary

Someone know a result of Sobolev Imbedding Theorem on manifold with boundary such that if $\Sigma $ is manilfod of dimension $k$ where we are know $W^{1,k}(\Sigma) \subset L^{2}(\Sigma)$ compactly, ...
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0answers
29 views

Differentiating a certain vector field

Let $M^n$ be a riemannian manifold and $p_0$ a point in $M$. Let $U$ be a normal neighbourhood of $p_0$, image of $B_{\delta}(0) = \{ x \in T_{p_0} M : \left\lvert x \right\rvert < \delta \}$ ...
0
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1answer
13 views

Checking a calculation with mean curvature and second fundamental form

Let $M$ be a 3 dimensional manifold, $N$ a surface in $M$ and $A$ the second fundamental form on $N$, $H$ the mean curvature. $h$ is the metric induced on $N$. I need to show that \begin{equation*} ...
1
vote
1answer
20 views

Smoothness from elliptic theory

I consider a smooth surface $N$ where the mean curvature $H$ is locally bounded and $N$ has locally uniform $C^1$ estimates. Then, a text states that " from elliptic theory, $N$ is smooth with ...
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0answers
22 views

Invariant under isometric

I am not familiar with the formal compute about the invariant under diffeomorphism (isometric),so I want a detail example. For example,$M,N$ are Riemannian manifolds, $\Phi :M\rightarrow N$ is ...
0
votes
1answer
34 views

Ricci soliton and Ricci flow

When I read the Ricci Soliton geometric meaning, I get stuck in the plugging in the Ricci flow as picture below.I don't know how to plug in it,in my opinion, Ricci flow is $\partial_tg_{ij}=-2R_{ij}$. ...
2
votes
0answers
27 views

Question about the normal bundle of a totally geodesic submanifold of a Cartan-Hadamard manifold

I'm reading the exposition in Lang's Fundamentals of Differential Geometry of the following generalization of the Cartan-Hadamard theorem: Suppose $X$ is a Cartan-Hadamard manifold (i.e. a complete, ...
2
votes
1answer
23 views

Inequalities for $L^2$ norms of gradients of functions that weakly converge in a Sobolev space

Let $\Sigma$ be a $k$-dimensional compact manifold with boundary. Suppose that $W^{1,k}(\Sigma) \subset L^2(\Sigma)$ is compact and that $\{\phi_j \}$ is a sequence that converges weakly in ...
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0answers
33 views

Do torsion and curvature have higher order analogues?

Consider the usual formulas for the Torsion and Curvature of an affine connection: $$T(X,Y)=\nabla_X Y -\nabla_Y X-[X,Y]$$ $$R(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]} $$ These formulas ...
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0answers
12 views

Approximations for an asymptotically flat manifold.

Let us consider an asymptotically flat manifold $M$ with the metric $g$. Let $\delta$ denote the flat metric and $p = g- \delta$. I am given the following approximations, where $C$ is a constant. 1) ...
0
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2answers
34 views

Example of a connected Riemannian manifold containing a non-compact closed bounded set

What is an example of a connected Riemannian manifold containing a non-compact closed bounded set? By the Hopf-Rinow Theorem, I know that the closed bounded sets of a connected Riemannian manifold ...
2
votes
1answer
48 views

Gradient vector field and level sets

So assume we have a complete Riemannian manifold $M$, and $f\in C^\infty(M)$. Suppose that $|\nabla f|=1$. Then if we let $p\in f^{-1}(0)$ does that imply that $f(\exp_p(t\nabla f))=t$. I asked an ...
2
votes
1answer
64 views

Riemannian manifolds isometry

Here is the following problem: Let $g_0$ be the Euclidean metric on $\mathbb C=\mathbb R^2$. Let $M=\{z \in \mathbb C| \ |z|<1 \}$ and equip it with the Riemannian metric ...
0
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1answer
35 views

Riemannian metric conformal to another metric

Suppose $M$ is a surface embedded in $\mathbb{R}^3$, then it has the natural induced Euclidean metric, denoted by $\textbf{g}$. Suppose $\tilde{\textbf{g}}$ is another Riemannian metric on $M$, we ...
1
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1answer
48 views

Solvability of Perelman's $\mathcal W$ system.

How to show the system have solution ? $R_{ij}$ is ricci tensor, $R$ is scalar curvature. I feel this is complex question, because I have little knowledge about PDE. So, if it is complex, just tell me ...
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0answers
16 views

Buseman function and isometry in Cheeger-Gromell splitting proof

So I have the Busemann function $b^+$ as in the proof of the Cheeger-Gromell splitting theorem in Peterson and I want to show that if I have the isometry $f:(b^+)^{-1}(0)\times\mathbb{R}\rightarrow M$ ...
0
votes
1answer
37 views

Killing fields and geodesic integral curves

My question is if you have a vector field whose integral curves are geodesic, does it imply that vector field is also killing? It seems like it is, just wanted to make sure if it was indeed true. In ...
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2answers
77 views

Why $\frac{d}{dt}\overline\lambda(g_{ij}(t))\ge \frac{d}{dt}(\mathcal F(g_{ij}(t),f(t))\cdot V^{2/n}(g_{ij}(t)))$?

$M$ is a Riemannian manifold,$g_{ij}(t)$ evolve under Ricci flow. $\lambda (g_{ij}) = \inf \{\mathcal F(g_{ij}, f) \mid f \in C^\infty (M), \int \limits _M \Bbb e^{-f} \Bbb d V = 1 \}$. ...
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0answers
31 views

Derivative of dual basis vectors in terms of Christoffel symbols

How can I demonstrate from $$ \frac{\partial \mathbf{e_j}}{\partial x^i} \equiv \Gamma_{ij}^k \mathbf{e_k} $$ what the value of $$ \frac{\partial \mathbf{e^j}}{\partial x^i} $$ (with the index now ...
0
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1answer
40 views

Cheeger-Gromoll splitting proof

Background information: $E_i$ is the parallel orthonormal frame along $c$ and $E_n=\nabla f\circ c$. Lemma: Let $M$ be a Riemannian manifold and $f\in C^\infty(M)$ with $||\text{grad} f||=1$. If $c$ ...
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0answers
18 views

If $M,N$ are (Riemannian) manifolds, $f: M \rightarrow N$ smooth, then what is $\frac{\partial}{\partial f^i}$? [duplicate]

Is this an element of the tangent bundle of $N$? I want to be able to write $$df(v^i \frac{\partial}{\partial x^i}) = v^i \frac{\partial f^j}{\partial x^i} \frac{\partial}{\partial f^j}$$ in local ...