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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Line in product mainifold

Let $(M_1, g_1)$ and $(M_2, g_2)$ be two complete Riemannian manifolds and consider the product $(M, g) = (M_1 \times M_2, g_1 + g_2)$. Let $\gamma : \mathbb{R} \to (M,g )$ be a line. I can write $t ...
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25 views

Obtaining embedding from geodesic

Suppose $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$. And suppose I know the induced Riemmanian-metric $g$ on $M$, which ...
3
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29 views

Understanding Energy minimization and poisson equation

Let $M$ be a Riemmanian manifold and $X$ be a vector field thereon. My question is why are these two problems equivalent?: \begin{equation} \operatorname{argmin}_{\phi}\int_M |\nabla \phi - X|^2 \...
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Optimizing potential

Let $X$ be a vector field on a Riemmanian manfiold $M$. I recently read that solving: \begin{equation} \operatorname{argmin}_{\phi}\int_M (\nabla \phi - X)^2 d\mu, \end{equation} where $\mu$ is ...
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How to derive the Christoffel symbols from the embeding view?

I encountered a statement in a book: If we define the manifold as embedded in a higher-dimensional Euclidian space, as a hypersurface $X(x)$, then the metric is given by $$g_{\mu\nu}=\partial_\mu X\...
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Metric non symmetric connection in the tangent bundle?

Let $\Sigma$ be a surface endowed with a Riemannian metric $g.$ According to the fundamental theorem of Riemannian geometry, there exist a unique $\nabla$ symmetric connection ( i.e. torsionless) in ...
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33 views

Covariant derivative in $\mathbb{R}^n$

I am studying my lecture notes on covariant derivative, and is having difficulty to do a computation: Suppose $X,Y$ be smooth vector fields in $\mathbb{R}^n.$ Consider the integral curve $c_p:(-\...
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19 views

Adjoint of the gauge covariant derivative

Suppose $A=A_1dx_1+A_2dx_2$ is a 1-form connection in $\mathbb{R}^2$ and $D_A \phi=d\phi-iA\phi$ is the gauge covariant derivative with $\phi=\phi_1+i\phi_2$ is a complex scalar field. May I ask what ...
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31 views

Can we define flat connection on any given smooth manifold?

For example, a sphere $S^2$ in $\mathbf{R}^3$ is apparently not flat with respect to the Euclidean connection, but can we define a flat connection and thus with affine charts on $S^2$?
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Critical point of Mabuchi energy functional has zero scalar curvature

I am currently reading the proof of convergence of Kähler-Ricci flow in the case $c_1(M)=0$ from Song, Weinkove. On page 45 he defines the term Mabuchi's $K$-energy functional: $$ \frac{d}{dt}\mathrm{...
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45 views

Reference about the surgery of Ricci flow

I roughly read the Topping's LECTURES ON THE RICCI FLOW. Seemly, there is not introduction about surgery. Seemly,it is enough to deal singularity by blow up. Then, for knowing surgery I read the ...
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19 views

Minimizing curve is differentiable

Suppose $M$ is a complete Riemannian manifold.Given two points $p$,$q$, there is a minimizing geodesic $\gamma$ connecting $p$and $q$ and the length of $\gamma=d(p,q)$. My question is, if a piecewise ...
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19 views

Every length minimizing $\mathcal C^1$ curve is a geodesic.

Let $(M,g)$ a manifold and $\gamma (t)$ for $t\in [a,b]$ a curve $\mathcal C^1$ the is minimizing the length. Then, if $p=\gamma (t_0)$ and $q=\gamma (t_1)$, then $\gamma $ is also minimizing the ...
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1answer
29 views

Geodesic sphere in $\mathbb H^2$

I saw the definition of a geodesic sphere, and I think I'm not able to "see" how do they look like. For example, it's obvious that in $\mathbb R^n$ geodesic spheres are simply normal spheres, and that ...
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158 views

Reference of what metric can be placed on manifold?

I just read some conclusion that $T^2$ can't be placed metric with positive curvature at all points. I don't know why is so . And what book introduce about this ? I mean about what metric can be ...
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28 views

Volume of a geodesic ball in a Riemannian manifold with $K<0$.

Let M be a simple connected Riemannian Manifold wih $K_M < 0$. Prove that the volume of any geodesic ball of M is strictly greater than $\frac{Vol(S^{n-1})r^n}{n}$, where $n = dim(M)$ and $r$ is ...
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First variation formula and geodesic

The first variation formula says that $$\left.\frac{\mathrm d }{\mathrm d s}\right|_{s=0}\ell(\varphi_s)=\frac{1}{c}\left[\left<\frac{\partial \varphi}{\partial s}(0,t),\frac{\partial \varphi}{\...
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1answer
37 views

Metric Matrix of the hyperbolic Riemannian manifold

Let $\Bbb{H}^n:=\left\{(x_1,...,x_n)\in\Bbb{R}^n\mid x_n>0\right\}$ be the hyperbolic space and $g={d^2x_1+\dots+d^2x_n \over x_n^2}$ be the standard hyperbolic metric. Looking at the $\left(\Bbb{...
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Geodesic of the half plan of Poincaré.

Don't worry, the question is short ! I just gave details on how things work, but there is nothing complicated On the half plan of Poincaré, we have the metric $$g=\frac{\mathrm d x^2+\mathrm d y^2}{y^...
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1answer
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If $f\in \mathbb C^\infty (M)$ and $(M,g)$ a riemanian metric, does $g(fX,Y)=fg(X,Y)$?

Let $(M,g)$ a riemanian manifold and $f\in \mathcal C^\infty (M)$. Does $$g(fX,Y)=fg(X,Y)\ \ ?$$ I know that $g$ is $\mathbb R-$bilinear, but is it also $\mathcal C^\infty (M)-$bilinear ? In fact, ...
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1answer
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Find the curvature of the sphere

I have to find the curvature (sectional, of Ricci and scalar) of $\mathbb S^n\subset \mathbb R^{n+1}$. My formula are For the sectional curvature: $$K_p(\pi)=\frac{R(X,Y,Y,X)}{\|X\wedge Y\|^2}$$ ...
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1answer
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Metric hyperbolic polar

The metric tensor hyperbolic is given by $$g=4\frac{\mathrm d x^2+\mathrm d y^2}{(1-x^2-y^2)^2}.$$ I have to right it in polar. I know that the euclidien metric $\mathrm d x^2+\mathrm d y^2$ is given ...
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1answer
13 views

If $\gamma _1$ and $\gamma _2$ are two different geodesic from $x$ to $y$, then, there is no minimal geodesic from $x$ to $z$

Let $(M,g)$ a Riemanian manifold and $x,y\in M$. Suppose there is two different geodesic $\gamma _1,\gamma _2$ that connect $x$ and $y$. Show that no one of these two geodesic are minimizing after $y$....
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Parallelizable open dense subset and integration

In Petersen's Riemannian Geometry (2016), it is stated on page 8 that any manifold $M^n$ has an open dense subset $O$ with $TO=O\times\Bbb R^n$. Thus it is orientable and one may define the integral ...
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1answer
59 views

Why the differential of exponential map is the identity.

Let $M$ a manifold and $T_pM$ it's tangent plan at $p$. We defined \begin{align*} \exp_p:U_p\subset \Omega _p&\longrightarrow M\\ V&\longmapsto \gamma _V(1) \end{align*} where $\gamma _V:I_V\...
3
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67 views

Why Differential Forms on Riemann surfaces?

I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces". Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:( It starts with the definition ...
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19 views

Geodesic flow generated by Riemannian distance function

This is an exercise in AC da Silva's Lectures onn Symplectic Geometry; I am having trouble showing the following. $(X,g)$ is a geodesically complete manifold, and $f: X \times X \to \mathbb{R}$ is ...
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38 views

Proof of iff condition of harmonic map

How to compute the equation above the red line in the picture below ? Below picture is from the Harmonic maps and their heat flows. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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1answer
36 views

how to define complete pseudo-riemannian manifold

In riemannin geometry, we define distance function by minimizing the length of curves. However we have nondefinite metric on psedu-riemannian manifold, so we cannot define a length of a curve as ...
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Flag Curvature in Finsler Geometry

Does anyone know what is the flag curvature in Finsler geometry? I looked for this definition, but I don't find any answer.
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1answer
26 views

About transformations of the metric: should we use the old or the new one to raise/lower indices?

Let $(M,g)$ be a (Pseudo-)Riemannian manifold. If I perform a transformation on the metric, getting a new metric $\tilde{g}$, which metric should I use to raise and lower indices? As I understand, ...
2
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1answer
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Confused About Indices in Deriving Curvature

Asking about a step regarding indices in deriving the Curvature tensor from the geodesic equation. Starting from $$ \frac{d v^a}{du} = - \Gamma^a_{bc}v^b \frac{dx^c}{du}$$ we integrate $$v^a(u) = ...
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existance of gradient conformal vector field on riemannian manifolds

Let ‎$ ‎(M,g)‎ $ ‎be a Riemannian manifold. A gradient conformal vector field on ‎$ ‎M‎ $‎ is a conformal vector field ‎$ ‎X‎ $‎ which is at the same time the gradient of a function on ‎‎$ ‎M‎ $ ‎:‎ \...
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1answer
25 views

Example of nonuniqueness of asymptotes of a ray

Let $(M, g)$ be a complete Riemannian manifold and let $\gamma : [0, \infty) \to \mathbb{R}$ be a ray, i.e. a unit speed geodesic such that for every $s, t \ge 0$ : $$ dist\big(\gamma(s), \gamma(t)\...
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1answer
42 views

Riemannian connection on Lie groups

Let $G$ be a Lie group with a bi-invariant metric. Then, the Riemannian connection is given by $\nabla_XY=\frac1 2 [X,Y]$ for all $X,Y\in \mathfrak g$. In the proof: Since $\langle X,Y\rangle$ is ...
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2answers
83 views

Distance function on complete Riemannian manifold.

Let $\left(M, g\right)$ be a complete Riemannian manifold. Let us fix a point $p \in M$ and consider the distance function $$ r(x) := \operatorname{dist}(x, p). $$ I would like to characterize the ...
5
votes
1answer
70 views

Is exponential map locally a diffeomorphism w.r.t. base point?

Let $M$ be a riemannian manifold and $\exp_p: T_pM \rightarrow M$ the exponential map at $p \in M$. At each point $p\in M$, $\exp_p$ can be restricted to a neighborhood $V$ of $0\in T_pM$ so that $\...
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1answer
29 views

Why the induced metric from the lie algebra of lie group $G$ is left invariant.

we say that a Riemannian metric on $G$ is left invariant if $<u,v>_y = <d(L_x)_y u,d(L_x)_y v>_{L_x(y)}$ to introduce a metric on $G$, take any arbitrary inner product $< , >_e$ on ...
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21 views

remannian connection \lamba_XX=0

$M$ is a Riemannian manifold with Riemannian connection $\nabla$. $X$ is a vector field on $M$. It is not in general true that $\nabla _XX=0$, for example a geodesic $\gamma$ satisfies $\nabla_{\frac{...
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1answer
50 views

Vanish christoffel symbols implies $g_{ij,k}=0$

Let $(M,g)$ be a pseudo-riemann manifold and $(U,\psi=(x^1,\ldots,x^n))$ a local chart around some point $p$ in $M$. It is easy to show that if $\partial g_{ij}/\partial x^k=0$ in $p$ for all $i,j,k$ ...
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Sectional curvature of 2-manifold

This is problem7, chapter 5 of Do Carmo's Riemannian geometry. Let $M$ be a Riemannian two manifold, $p\in M$ , $exp_p$ is a diffeomorphism on a neighbourhood of origin $V\in T_pM$. Let $S_r(0)\...
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What will happen if evolve metric under Ricci flow on general manifold? [closed]

Because the scalar curvature under Ricci flow evolve by $$ \partial_t R=\Delta R+ 2|Ric|^2 $$ I treat it as heat equation with heat source . So, no matter the heat source is very hot or not , the ...
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1answer
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normal connection on immersed hypersurface vanishing

I am studying Riemannian geometry using Do Carmo's book. I am learning about isometric immersions right now, and I got stuck with the following claim about Codazzi's equation. Let $f:M^n \to \...
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1answer
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Connection after a metric rescaling and compatibility

It's known (see here for example) that after a rescaling of the metric $\tilde{g}=e^{2\omega}g$, we can find a new connection $\tilde\nabla$ associated to the new metric: $ \tilde\nabla _X Y = \nabla ...
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Convergence of Discretized Geodesics?

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$. Suppose $f^{-1}:U_p \mapsto \mathbb{R}^D$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the ...
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1answer
50 views

Lie bracket and inner product

$X, Y, N$ are vector fields on a riemannian manifold $M$. $\langle X, N \rangle(p)=0 $, $\langle Y, N \rangle(p)=0$ at some point $p$. Show $\langle [X,Y], N \rangle(p)=0$. I want to use $[X,Y]=XY-...
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2answers
71 views

Problem to conceptualize $\frac{\partial }{\partial \theta}$ and $\frac{\partial }{\partial \varphi}$.

I have some little problem to give a conception to $\frac{\partial }{\partial \theta}$ and $\frac{\partial }{\partial \varphi}$ on manifold (like $\frac{\partial }{\partial x}$ as well). For example, ...
2
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1answer
38 views

Curvature of the sphere, how can I get it?

I want to compute the curvature of the sphere. I have the following definition : The curvature is given by $$K_p(T_p\mathbb S^2)=\frac{R(X,Y,Y,X)}{\|X\wedge Y\|^2}$$ where $X,Y$ is a basis of $T_p(\...
2
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1answer
54 views

Hodge theory on semi-Riemannian manifolds [reference request]

I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the ...
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27 views

the map from the horizontal bundle is a submersion or an immersion

Let $\pi \colon M \to B$ be a riemannian submersion and $g$ the metric on $M$. Then we get the vertical subspace $ \mathcal{V}_x = \ker d_x \pi$ and the horizontal subspace $\mathcal{H}_x= \mathcal{V}...