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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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 ...
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49 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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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 $f:(b^+)^{-1}(0)\times\mathbb{R}\rightarrow M$ ...
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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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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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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 ...
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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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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 ...
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32 views

Smooth extension of a tangent vector

Let $(M,g)$ be an $n$-dimensional smooth Riemannian manifold and $p_0\in M$, $v_0\in T_{p_0}M$ with $|v_0|=1$. $\nabla$ is the Levi Civita connection. How can we construct a smooth vector field $X$ ...
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36 views

Cheeger-Gromoll Splitting in Peterson's text

So I don't see how the distance function allows us to conclude that $M=U_0\times \mathbb{R}$. Could someone explain it in more detail.
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61 views

Book recommendation on PDE

I want to read some book about PDE, especially about parabolic differential equations. I know a little basic conception and theorem of PDE,and it's very basic and little.So, I hope it is not too ...
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45 views

Are there more convenient charts than Riemannian normal coordinate chart?

Let $(M,g)$ be an arbitrary smooth Riemannian manifold of $n$ dimensional and $p_0\in M$. $\nabla$ denotes the Levi Civita connection. It is well known that there is a coordinate chart around $p_0$ ...
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29 views

Strong maximum principle on Riemannian manifolds

So I am unsure of how the hessian becing less than or equal zero contradicts that the laplacian is greater than zero. Can someone provide some input for the reasoning. Thanks.
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17 views

Lie derivative of Killing form

One may choose the Killing form $Tr(T^aT^b)$ as the metric $g$ on a Lie group $G$. It is known that the Killing form is invariant under the adjoint transformation, i.e., $\delta Tr(XY)= ...
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119 views

If the Exponential map is a diffeomorphism at a point, can we say something about other points?

Let $M$ be a complete (connected) Riemannian manifold, $p \in M$ some point in $M$. Assume $exp_p$ is a diffeomorphism from $T_pM$ onto $M$. Is it true that $exp_q$ is a diffeomorphism for all ...
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35 views

Vector invariant under the flow

I'm studying a paper by Demir N. Kupeli, On submanifolds in spacetimes, and during a proof of a proposition, the author say: "Extend $X\in T_p S$ by making it invariant under the flow generated by ...
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Lemma 40(Maximum principle) in Peterson

So let me state the lemma first Lemma 40:If $f,h:(M,g)\rightarrow \mathbb{R}$ are $C^2$ functions such that $f(p)=h(p)$ and $f(x)\geq h(x)$ for all $x$ near $p$, then $$ \begin{align} \nabla ...
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29 views

Does a coordinate basis exist locally on any manifold?

A holonomic or coordinate basis for a differentiable manifold is a set of basis vector fields $\{e_k\}$ such that some coordinate system $\{x_k\}$ exists such that $e_k=\partial/\partial x_k$. A ...
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21 views

Structure of $C^k$ ($k<\infty$)Riemannian metrics on a manifold

$M$ is a smooth manifold. It's known that if $M$ is compact, then the space of smooth Riemannian metrics has a Frechet manifold structure. For the space of $C^k$($k<\infty$) Riemannian metrics, ...
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26 views

Expanding breather and expanding gradient soliton

As picture below, the proof of $(i)$ of corollary $1.5.5$,I can't find why the monotonicity is strict unless we are on a gradient expanding soliton. I know the gradient expanding soliton as $\exists ...
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64 views

How to show that $\frac{1}{V}\int_MR \, dV\ge\mathcal{F(g_{ij},f)}$?

When I read the proof of Corollary 1.5.5 of this paper (204th page),I get stuck in the red box in picture below.How to show it ? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ...
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27 views

Huisken's Distance Comparison principle for Ricci flow

So I have been studying curvature shortening flows, and I was curious if there were some variants of Huisken's distance comparison principle for Mean curvature flows or even the Ricci flow. It seems ...
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40 views

Show that $\int_M\Delta f HdV=-\int_M\nabla f\cdot\nabla H dV $?

$M$ is a compact Riemannian manifold,$f,H$ are functions on $M$,How to show that $\int_M\Delta f HdV=-\int_M\nabla f\cdot\nabla H dV $?Besides,if $M$ is not compact, is the equality right? Beside, I ...
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109 views

Beginner's book for Riemannian geometry

Could you recommend some beginner books for Riemannian geometry to me? I am completely new to Riemannian geometry, but have some basic knowledge of differential geometry. I am looking for a book in ...
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39 views

Relation of breather and gradient soliton.

As picture below , I am fuzzy with the red line. As definition 1.5.1, assuming $g_{ij}(t)$ is expanding breather,then for some $t_1<t_2 \text{ and }\alpha>0$,$\alpha g_{ij}(t)$ and $g_{ij}(t)$ ...
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31 views

Is the cut locus geodesic?

Let $P$ be a point on compact Riemannian manifold $M$. Let $L$ be the cut locus. Let $Q$ be a smooth point of $L$. Is $L$ totally geodesic in a neighborhood of $Q$?
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38 views

First eigenvalue in Ricci flow.

In the picture below, why is $\lambda(g_{ij})$ the first eigenvalue of $-4\Delta+R$? The first eigenvalue is strange to me. After reading the wiki about it, I find $$\lambda_1=\inf\limits_{u\ne ...
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42 views

Where is $-2R_{ij}\nabla_jf$ from?

$M$ is a compact Riemannian manifold,$g_{ij}$ and $f(t)$ is defined as first picture. I want to compute (as equality with red line in second picture) $$ \int_M-\nabla_if\nabla_i(2\Delta ...
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62 views

Nonexistence of conjugate points $\Rightarrow$ a geodesic is minimizing

Motivation: I am trying to prove a certian geodesic is minimizing. The only generic tool I know for doing that is the fact the a gedoesic is minimizing as long as it stays in a normal neighbourhood of ...
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48 views

Isometry and integral over manifold

Let $(M,g)$ be a Riemannian manifold, $(U,\varphi), (V,\psi)$ be two charts and $\phi:U\rightarrow V$ be some isometry. I would like to prove the formula $$\int_Vf dV_g=\int_U(f\circ\phi)dVg$$ I ...
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49 views

How to show that $\int_M(\Delta f-|\nabla f|^2)(2\Delta f-|\nabla f|^2)e^{-f} dV=\int_M-\nabla_if\nabla_i(2\Delta f-|\nabla f|^2)e^{-f}dV$

$M$ is a compact Riemannian manifold. $f$ is function on $M$ How to show that $\int_M(\Delta f-|\nabla f|^2)(2\Delta f-|\nabla f|^2)e^{-f} dV=\int_M-\nabla_if\nabla_i(2\Delta f-|\nabla ...
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What is the mean of $R(v_{ij})$

As picture below ,I can compute $\delta R=-v_{jl}R_{jl}+\nabla_i\nabla_lv_{il}-\Delta v$.But in the last line (1.5.2), what is the mean of $R(v_{ij})$? I think $R$ is a function on manifold, why the ...
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83 views

Tangent spaces, how are vectors parallel transported?

I understand that tangent vectors lie in separate tangent spaces based on the point on which they are tangent to a manifold, but what about vectors that are parallel transported? For any manifold ...
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51 views

Geodesic axis and displacement function

So let us assume our manifold is complete, simply connected, and has nonpositive sectional curvature. If we assume that the displacement function $f(x)=d(x,\phi(x))$, for an isometry ...
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Affine Kac-Moody Group Isometry of a Manifold

An isometry of a Riemannian manifold is an infinitesimal displacement generated by a Killing vector field $V=\zeta^aV_a=\zeta^aV_a^i\frac{\partial}{\partial x^i}$. If the isometry corresponds to the ...
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Integral curves and null geodesics

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p ...
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52 views

Is a Riemannian manifold with isometric coordinate charts flat?

Suppose $M$ is a Riemannian Manifold such that each point has a coordinate chart into $\mathbb{R}^n$ that is an isometry, in the sense that the inner products are preserved. Does this imply that $M$ ...
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45 views

Relation between curvature and sectional curvature

Let $(M,g)$ be a Riemannian manifold and $ h = c.g$ for some $c > 0$ . Then the Levi-Civita connections of $g$ and $h$ are same. From the above deduce the relation between corresponding curvature ...
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25 views

Comparison triangles on manifolds and Euclidian arguments

So recently I've learned about the comparison theorems and I have some questions. I know that when we construct comparison triangles were able to carry over some properties we know about Euclidian ...
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1answer
43 views

Computing the Lie derivative of a metric

Suppose $g$ is a Riemannian metric on a manifold, and $X$ is a smooth vector field. Show that $$(\mathcal{L}_Xg)_{ij} = \nabla_iX_j + \nabla_jX_i$$ I've been able to obtain that ...
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96 views

Compute $\frac{\partial}{\partial t}\mathcal{F}(g_{ij},f)$

How to get the equality 1 ? When I compute it ,I get stuck in $\frac{\partial}{\partial t}R_{ij}$ and $\frac{\partial}{\partial t}(\nabla_if\nabla_jf)$. I don't know how to deal the two terms. Below ...
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whether $t=0$ is missed in the proof?

When I try to compute $g^{ij}g^{pq}\nabla_{i,j}^2h_{pq}=\Delta tr_gh$, I need $g^{ij}=\delta^{ij},\nabla_i g_{kl}=0$,i.e, it's normal coordinate. But $g(t)$ will change with $t$, only for a $t_0$, I ...
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18 views

Differential of function on Hilbert space or Banach space .

In the picture below, I think the life part of 9.1 is something like directional derivative. But in the right part , I don't know what $dE(u)$ is ,I guess it should be a function, but when $E: ...
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28 views

Show that $R(\partial_i,\partial_j)h_{kp}=0$.

In picture below, I am not sure why $R(\partial_i,\partial_j)h_{kp}=0$. I think it is because we can consider $h_{kp}$ as a function. Am I right ? The picture below is from 57th page of this paper. ...
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35 views

Induce volume form

In picture below,$\iota$ is inclusion.I guess $v$ is outer normal vector. Why the volume form is a vector times $dx$ ? I mean that the Riemannian volume form is $\sqrt{det(g)}dx_1...dx_n$ , is a ...
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39 views

Compare Parallel Transport in a Submanifold to Geodesic Parallel Transport in Ambient Manifold

Suppose I have a smooth Riemannian manifold $(\tilde{M},g)$ and an embedded submanifold $M \subset \tilde{M}$. I fix a point $p \in M$ and $X_p \in T_pM$. Suppose that $q \in M$ is joined to $p$ by ...
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Ricci identity in Lectures on Ricci Flow

So in the book, Lectures on Ricci flow, the identity is given as $$-\nabla^2_{X,Y}A(W,Z,\ldots)+\nabla_{Y,X}^2A(W,Z,\ldots)=-A(R(X,Y)W,Z,\ldots)-A(W,R(X,Y)Z,\ldots)$$ where ...
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22 views

Smoothness of Ricci flow

Consider the solution $g_t$ to the Ricci flow equation $\frac{\partial g}{\partial t} = -2\text{Ric }g$ on a compact manifold $M$ with initial metric $g_0$. This is probably very elementary, but I was ...
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22 views

Isometry invariance of Christoffel symbols

An isometry of a Riemannian manifold is generated by a Killing vector field $X$ with Lie derivative of the metric $L_X g=0$. Does this immediately imply that the Lie derivatives of the Christoffel ...
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23 views

Existence a path of a smooth manifold

Given a continuous differentiable functio $F:\mathbb{R}^n\mapsto \mathbb{R}^m$ with $n>m$. Define $$ {\cal M}=\{x\in\mathbb{R}^n: F(x)=0\} $$ and let $x_0\in{\cal M}$ such that the Jacobian of ...