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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Gradient flow under varying riemannian metric

Suppose I have a smooth flow $\varphi(t)$ on some Riemannian manifold $(M,g)$ and I know that $\dot{\varphi}(t) = \textrm{grad }F$ for some smooth function $F$. If I smoothly modify the metric $g$, ...
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Reference request-(Criterion for a manifold to be real analytic)

Let $(M , g)$ be a Riemannian manifold. If there exists a subatlas of normal coordinate systems such that the $g_{ij}$ are real analytic functions with respect to each normal coordinate system in the ...
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Lie derivative of metric $g$

Let $\varphi_t$ be a one-parameter group of diffeomorphisms generated by a vector field $V$ on $M$, and the metric is given by $$ g_{ij}(x,t)=\varphi_t^*g_{ij}(x,0) $$ How to show that $\partial_tg=\...
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Computing explicitly the Riemannian Distance on $GL_n^+$

$\newcommand{\ep}{\epsilon}$ Let $GL_n^+$ be the Lie group of invertible $n \times n$ matrices with positive determinant. In particular it's a connected open submanifold of the Euclidean space $\...
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Show that $\operatorname {trace}(X\rightarrow R(X,Y)Z)=\sum_{i=1}^n\langle R(e_i,Y)Z,e_i\rangle$

$\langle\,,\rangle$ is Riemannian inner,and R(X,Y)Z is curvature tensor. How to show that $\operatorname{trace}(X\rightarrow R(X,Y)Z)=\sum_{i=1}^n\langle R(e_i,Y)Z,e_i\rangle$ ?
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Riemannian metric under coordinate exchange

Let $\{x^i\},\{y^a\}$ be coordinate functions on a common open set.why we have $g_{ab}(y)=g_{ij}(x)\frac{\partial x^i}{\partial y^a}\frac{\partial x^j}{\partial y^b}$ ? And $g^{ab}(y)=?$
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Local expression of Lie bracket

Let $M$ is a Riemannian manifold, $\forall X,Y\in T_pM$ and $\forall f$ is a function on $M$,we have $$ [X,Y]f=X(Yf)-Y(Xf) $$ If $\{x^i\}$ is a local coordinate of some neighborhood of $p$,$X=X^i\...
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A question about Gauss's Lemma in Riemannian Geometry.

I was reading the proof of Gauss Lemma from do Carmo's book on Riemannian geometry. We need to prove $\langle (d\exp_p)_v(v), (d\exp_p)_v(w) \rangle = \langle v,w\rangle$. He decomposes $w=w_T+w_N$ ...
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Compute Hessian for a given line element

Given a $t,r$ coordinate system with line element $ds^2=Edt^2+Gdr^2$, i have to compute $H^r(\partial_t, \partial_t) = E_r/2G$. I tried using the definiton of the Hessian; $H^r(\partial_t, \partial_t)...
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33 views

Tangent mapping under the local coordinate

As we know ,if $\varphi:M\rightarrow N$ is a differentiable map, then ,the map $\varphi$ will induces a linear map $\varphi_*:T_xM\rightarrow T_{\varphi(x)}N$ ,up to $(\varphi_*X)(f)=X(f\circ\varphi)$,...
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Mean value operator on Riemannian manifold

Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$) Consider the mean value operator, ...
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What property does Gauss-Bonnet theorem indicate for a function defined on a 2D Riemannian surface?

If I have a function defined on a 2D Riemannian surface, what can Gauss-Bonnet theorem tell me about the property of this function? I just want to get a picture of this theorem in my mind. An example ...
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Ricci curvature under Ricci flow

In fact ,I got stuck in some compute. Let $B_{ijkl}=g^{pr}g^{qs}R_{piqj}R_{rksl}$, in the below picture,I can't compute out the 1 and 2 equation above red line. In the first, I know how to get $\...
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40 views

Show that $\partial_t g^{jl}=-g^{jp}(\partial_tg_{pq})g^{ql}$

If $g_{ij}$ are the components of a Riemannian metric and $\partial_tg_{ij}=-2R_{ij}$, how to show $$\partial_t g^{jl}=-g^{jp}(\partial_tg_{pq})g^{ql}\,?$$
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Curve on Lorentz manifold

If $\alpha : [0,B) \rightarrow M, B \leq \infty$, is an extendible, piecewise smooth (nonspacelike curve) in a Lorentz manifold, then $\alpha$ has a finite length. Any hints on how to show this? I ...
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26 views

Solving a PDE Using Variational Calculus

I am currently trying to understand a proof of the following Claim Let $(\mathcal{M},g)$ be a compact, oriented Riemannian manifold and let $f:\mathcal{M} \to \mathbb{R}$ be a function. Then $$ \...
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Exchange formula of 2-form or n-form

I only know the exchange formula of 1-form is that $\nabla_i\nabla_j v_kdx^k-\nabla_j\nabla_iv_kdx^k=g^{lm}R_{ijkl}v_mdx^m$. But in the below picture, only exchange formula of 1-form seems be not ...
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Why the $\partial_t(\Gamma_{ip}^k \Gamma_{jl}^p -\Gamma_{jp}^k\Gamma_{il}^p)$ vanish?

In the red line part of below picture ,why the $\partial_t(\Gamma_{ip}^k \Gamma_{jl}^p -\Gamma_{jp}^k\Gamma_{il}^p)$ vanish ?I know $\Gamma$ will vanish under normal coordinate. But if so, the RHS ...
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Compute $\partial_t \Gamma_{ij}^k$ under Ricci flow.

The $g_{ij}$ is Riemannian metric ,and is solution of $\frac{\partial }{\partial t}g_{ij}(x,t)=-2R_{ij}(x,t)$. In the below picture , computing the $\frac{\partial}{\partial t}\Gamma_{jl}^h$. I ...
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A lemma for proving Myers-Steenrod theorem

So here is what I am having problems with. The lemma is that if we suppose $(M,g)$ is a Riemannian manifold, and let $p\in M$. Then if $X,Y\in T_pM$ we have the following $$\lim_{t\rightarrow\infty}\...
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Is the principal curvature of a cylinder positive or negative according to the second fundamental form?

First off, what is the name of the tensor associated with the second fundamental form? For the first fundamental form, I believe we call the associated tensor, "the metric tensor." Principal ...
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Riemannian & metrics

It's well known that a connected Riemannian manifold induces a metric space: the distance between two points ia measured as the infimum of the length of curves joining the points. The inverse: given a ...
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Coefficients of connection under diffeomorphism and metric change.

Let $\varphi_t :M\rightarrow M $ is a family of diffeomorphism. $\widehat{g}_{ij}(x,t)$ is a solution of $$\frac{\partial}{\partial t}g_{ij}=-2R_{ij} ,\ y(x,t)=\varphi_t(x)=\{y^1(x,t),...,y^n(x,t)\},\...
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44 views

Why $\Delta f-|\nabla f|^2 =0$ ,$f$ is a function on compact manifold.

There is a explain in the below picture ,but I don't know how to compute the 1.1.17. And how to compute the integrate of 1.1.16 ?
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35 views

Shrink and expand of homothetic gradient Ricci soliton

For a homothetic gradient Ricci soliton, $$ R_{ij}+\nabla_i\nabla_jf-\lambda g_{ij}=0 $$ Why for $\lambda>0$ the soliton is shrinking? Why for $\lambda <0$ it is expanding ?
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Compute of exchange of covariant derivatives.

Let $v=v_kdx^k$,$R_{ijkl}=g_{kp}R^p_{ijl}$,How to show that : $$ \nabla_i\nabla_jv_k-\nabla_j\nabla_iv_k=R_{ijkl}g^{lm}v_m $$
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25 views

The Riemannian Distance function does not change if we use $C^1$ paths?

In an answer to this question, it is proved that the Riemann distance function is the same whether we use smooth paths or piecewise smooth paths. What happens if we use $C^1$ paths? Can we ...
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proving a particular subset of a Riemannian manifold is closed using continuity

I have two Riemannian manifolds, $(M,g)$ and $(\widetilde{M},\widetilde{g})$ and two maps $\varphi, \psi : M \to \widetilde{M}$, which are both local isometries. I am trying to show that the set $A = ...
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Compute of metric and curvature under transformation of coordinates.

Let $\widehat{g}_{ij}(x,t)$ be a solution of $$\frac{\partial g_{ij}}{\partial t}=-2R_{ij},$$ and $\varphi_t:M\rightarrow M$ is a family of diffeomorphisms of $M$. Let $$g_{ij}(x,t)=\varphi_t^*\...
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Symbol of operator and eigenvectors of symbol

In my opinion ,the symbol of operator is a polynomial that $\partial_i$ be replaced by $\xi_i$. And on the Wiki,there is some examples that the operator acts on one function $u$. But in the below ...
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The linearization of a system and the derivative of operator.

Firstly, in red line 1 of the picture below,whether it means make a variable substitution $\widetilde {g}_{ij}=f(x)g_{ij}$? Because in my opinion, the linearization of something is to make a ...
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Definition of Riemannian Metric

Let $(M, g)$ be a Riemannian manifold. Standard definitions of a Riemannian metric $g$ states that $g$ specifies a symmetric, bilinear, positive definite form on each tangent space $T_{p}M$ that ...
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Geodesics on the set of disordered points of $R^d$

My problem is the following: I consider the set of unordered configurations of $\mathbb{R}^d$ as the quotient of the manifold $F(\mathbb{R}^d, n) = \lbrace (x_1, \dotsc, x_n) \in (\mathbb{R}^d)^n ; \...
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Sign of Riemannian and application of commutating formula .

$R_{ijkl}$ is curvature ,$\nabla $ is Riemannian connection ,$f$ is a function on Riemannian manifold. $g^{ij}$ is inverse of $g_{ij}$. If $\nabla_i\nabla_jv_k-\nabla_j\nabla_iv_k=R_{ijkl}g^{lm}v_m$...
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How to compute that $\mathcal{L}_Vg_{ij}=g_{ik}\nabla_jV^k+g_{jk}\nabla_iV^k$

How to compute that $\mathcal{L}_Vg_{ij}=g_{ik}\nabla_jV^k+g_{jk}\nabla_iV^k$ ? The $g_{ij}$ is Riemannian metric, $V=V^k\frac{\partial}{\partial x^k}$ is vector flied ,$\mathcal{L}_Vg$ is Lie ...
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starry regular icosahedron

La estructura del pentágono regular estrellado se puede describir mediante una curva f(t) que no se corta a sí misma en una superficie de Riemann de dos hojas. Cómo onstruir en una variedad ...
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50 views

Commuting Covariant Derivatives in Derivation of First Variation Formula

I'm following the book "A Course in Minimal Surfaces" by Colding and Minicozzi. I'm stuck on section 1.3, The first variation formula. We are given a Riemannian manifold $M$ with metric $g$ and ...
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proving a given curve is a geodesic

I am trying to solve the following problem from Lee's Riemannian Manifolds: where the curve $\gamma : I \to \mathbb{R}^2$ is given by $\gamma(t) = (a(t),b(t))$ so that $M$ is parametrized as $\...
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Are Christoffel symbols invariant under reparameterization of the curve?

Let be $M$ a Riemannian manifold with Levi Civita connection. In the equation of geodesics appears the term $$\Gamma^k_{j} \circ \sigma(t) ,$$ where $\sigma :I \rightarrow M$ is a curve and $\Gamma^...
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Definition of the smooth local convergence of a sequence of hypersurfaces

I'm reading a lecture note on mean curvature flows (by C. Mantegazza) and studying about analysis of type-I singularities. Let $M$ be a smooth closed $n$-dimensional manifold and $\varphi:M\times[a,\...
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Parallel transport convergence

So I was working on a problem and I got stuck. The question is as follows. Let $\pi:E\rightarrow M$ be a vector bundle of rank n with a metric and a metric connection on E. Now suppose that we have a ...
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Invariant for the action in SE(2)

The action of $SE(2) = SO(2) \ltimes R^2$ on smooth curves in the plane is definition $(R_{\theta},(a,b)) \cdot (x,u(x))=R_{\theta} \cdot (x,u(x))+(a,b)$  I have already shown that $\frac{u_{xx}}{(...
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Show that the metric $d$ on a Riemannian manifold can be defined using $C^\infty$ instead of piecewise $C^\infty$ curves

I have seen the usual definition of the distance in a Riemannian manifold $M$, considering piecewise $C^\infty$ curves. I know that we consider this kind of curves only for convenience (for example ...
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A question about a function used to define the length of a curve in Riemannian geometry

Let be $(M,g)$ a connected Riemannian manifold. If $ \phi : [a,b] \rightarrow M$ is $C^\infty$ we define the arc-length of the curve $\phi$ as the quantity: $$J(\phi )= \int_a^b f(\phi (t),\dot{\...
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What does the compatibility condition in the definition of meromorphic differentials mean?

Let $S$ be a Riemann surface, with an atlas $(U_i, \varphi_i)_{i \in I}$. For any $P \in S$, denote $$\frac{dz_i}{dz_j}(P):= (\varphi_i \circ \varphi_j^{-1})^\prime (\varphi_j(P)).$$ We then define a ...
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Isometric embedding of a hemisphere of $\mathbb{S}^n$

Herman Karcher says at the end of the first page in Riemannian Center of Mass and so called karcher mean that for the Hiperbolic space $\mathbb{H}^n$ and a hemisphere of $\mathbb{S}^n$ there are ...
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Is it true that a geodesic on a hyperbolic surface can be lifted to a geodesic on the hyperbolic plane?

Let $\mathbb{H}$ be the hyperbolic plane, $\Gamma < \text{Isom}(\mathbb{H})$ be a Fuchsian group, and $S = \mathbb{H}/\Gamma$. If $\gamma : [0,1] \rightarrow S$ is a geodesic, can it be lifted to a ...
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Existence of a free “sub-isometric” embedding of a Riemannian manifold

I am trying to understand Matthias Günther's proof of Nash's embedding theorem which is outlined in Günther, Matthias. Isometric embeddings of Riemannian manifolds. Proceedings of the International ...
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1answer
30 views

Why $-2Ric=\mathcal{L}_vg$?

$\varphi_t$ is a one-parameter group of diffeomorphisms generated by a vector field $V$ on $M$, $$ g_{ij}=\varphi_t^*g_{ij}(x,0) ~~~ \frac{\partial g_{ij}}{\partial t}=-2R_{ij} $$ How to show that $-...
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69 views

Convexity and mean curvature

Let $N$ be a Riemannian 3-Manifold with and $M \subset N$ an embedded, oriented codimension $0$ submanifold-with-boundary, bounded by a non-empty smooth subsurface $S := \partial M$. Now, with $M$ and ...