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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How to define gradient of an affine connection

I heard somewhere (and just read on a physics forum) that the gradient of a smooth function $f$ on a manifold $M$ can be defined when $M$ is equipped with an affine connection on its tangent bundle, ...
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Is this computation of the Christoffel coefficients on a Kähler manifold correct?

Let $M$ be a Kähler manifold (in truth, I am only interested in $\Bbb C \Bbb P^n$). Is it possible to express the Christoffel coefficients of the Levi-Civita connection in terms of the coefficients of ...
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12 views

Product of Riemannian manifolds and volume element

Let $X$ and $Y$ be Riemannian manifolds and consider a function \begin{align} f\colon X\times Y &\to \mathbb{R},\\ (x,y) &\mapsto f(x,y) \end{align} Now I have to integrate $f$ with ...
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Levi-Civita connection with biinvariant metric

I'm struggling with the proof of the following, well-known result for the Levi-Civita connection of a Lie Group with biinvariant metric, i.e. satisfying \begin{equation} g(D_bL_a X_b, D_b L_a Y_b)= ...
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Variational formulation of harmonicity on Riemannian manifolds

$\newcommand{\R}{\mathbb{R}}$ I am trying to follow a derivation of the first variation formula for the energy functional. (In "Selected Topics in Harmonic maps"). Here is the context: $M,N$ are ...
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Isometries of Riemannian manifolds are harmonic?

Let $(M,g),(N,h)$ be two Riemannian manifolds. Assume $f:M \to N$ is an isometric immersion. Is $f$ harmonic? (i.e a critical point of the energy functional) (I know this is true when ...
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61 views

Proving a compact Lie group admits a biinvariant metric [duplicate]

At the end of a lesson in Differential Geometry, my teacher said: Fatto, che non dimostriamo, non è difficile ma il tempo scarseggia, se $G$ è compatto possiamo sempre trovare una metrica ...
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14 views

Parametrizations and notation

I am studying riemannian geometry using Do Carmo´s book. The first thing I have to understand (later) is some terminology/ notation about parametrization. In the chapter 0 states this (after defining ...
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When is the metric completion of a Riemannian manifold a manifold with boundary?

Let $(M,g)$ be a connected smooth Riemannian manifold and denote by $(M,d)$ the induced metric space following by taking topological metric to be the infimum over length of curves in the standard way. ...
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Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: ...
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Physical meaning of Hawking's Singularity theorem

I'm studying O'Neill's "Semi-Riemannian Geometry with applications to Relativity". I know that the following theorems are related to the Big Bang, but I don't understand how. Let $M$ be a ...
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The Hawking mass is non-decreasing during “jumps” (Penrose Inequality)

I am reading $\textit{The Penrose Inequality}$ by H. Bray and P. Chrusciel and I am stuck at one of their statement. The question regards the fact that the Hawking mass is non-decreasing during a ...
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22 views

lower bound on volume of balls

It is well known that a lower bound on Ricci curvature gives an upper bound on the volume of balls. What are conditions that gives a lower bound on the volume of balls? It is reasonable to think that ...
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30 views

How does one determine whether a coordinate basis is orthogonal or not?

Apologies for what is perhaps a very basic question, but I have been studying differential geometry with a view to gain a deeper understanding of general relativity and I have hit a stumbling block. ...
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Is any smooth deformation of a metric in dimension 1 conformal?

Consider $(S^1, g)$ where $S^1$ is the unit circle and g is a metric. Now consider the metric $$ \tilde g := f g $$ where f is a smooth positive function. Since in 1 dimension this is the only smooth ...
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14 views

Conformally flat manifolds

Let $(\mathbb{R}^n,e^{\lambda}g_e)$ be a conformally flat manifold with constant sectional curvature. Is it true that we think of the conformal factor $e^{\lambda}$ is a constant function on ...
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28 views

A step in proof of Bishop-Gromov Theorem

I am reading the proof of Bishop-Gromov's comparison theorem in Schoen and Yau's Differential Geometry book. They do it using the Jacobi fields approach. There is a step which I have trouble ...
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20 views

Finding a domain of an integral curve of a vector field

Studying Morse theory, I am stuck on some problem. Let $M$ be a compact smooth manifold, and $f$ is a smooth real-valued function on $M$. Choose a Riemannian metric $g$ on $M$, let $X$ be the vector ...
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1answer
24 views

Christoffel symbols of $S^n$ in polar coordinates

Consider the usual local polar coordinates $\theta_1, \theta_2,..., \theta_n$ on $S^n$. We were taught about Christoffel symbols today and I am trying to see what the Christoffel symbols of $S^n$ ...
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22 views

Hessian of the Stereographic projection

Consider the stereographic projection from the sphere $S^n$ onto $\mathbb{R}^n$, and take the usual local spherical (polar) coordinates $\omega_1,..,\omega_n$ on $S^n$ (coming from its embedding in ...
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Is the Riemann Hypothesis incorrect? [closed]

See the attached image I would like to know your opinion about if the zeros shown in this picture can be considered as the zeros mentioned by Riemann in his Z function. I think yes and that his ...
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1answer
21 views

Why is $\nabla_X (\varphi Y)=\nabla_X(0\cdot\varphi Y)$?

When I read Lee's Riemannian Manifolds : An Introduction to Curvature, I am confused by the red line in the picture below. Why is $\nabla_X (\varphi Y)=\nabla_X(0\cdot\varphi Y)$?
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Harmonic map into $S^n \times \mathbb{R}$

Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) ...
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1answer
30 views

Understanding the first fundamental form of a surface, how the parametrization doesn't matter.

The following is an excerpt from Pressley's Elementary Differential Geometry on the definition of the first fundamental form. However, there are some parts of this concept that I'm unclear about. It ...
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46 views

This map is an isometry (in the Riemannian sense) of the hyperbolic plane. Why is the following a proof of it?

I'm making my way through a textbook on elementary undergraduate geometry. The author has defined the notion of an isometry between two subsets of $\mathbb{R}^2$ equipped with a Riemannian metric. It ...
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2answers
54 views

Parallel Translation is Path Independent iff Manifold is Flat

Problem. Let $M$ be a smooth Riemannian manifold and $\nabla$ be the Levi-Civita connection. Then the following are equivalent $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z\equiv 0$ ...
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1answer
22 views

Left multiplication isometry?

If $G$ is a semi-simple Lie group and $g\in G$, then $G$ has a bi-invariant metric which is a Riemmanian metric. My question is: with respect to this metric does the left-translation map ...
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16 views

Integration over two sub manifolds

I have two integrals which I would like to evaluate, but I am not sure if my formulas are correct. Consider a two dimensional closed Riemannian manifold $(M,g)$. I would like to calculate two ...
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33 views

Harmonic map and pullback metric

Let $\phi : M \to \mathbb{R}^n$ be a harmonic map, where $M$ is a Riemannian manifold. Let us take coordinates $(u_1, u_2,..., u_n)$ on $\mathbb{R}^n$ and express the Euclidean metric as $g = \Sigma ...
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28 views

Change of coordinates in target space of map

Consider a function $\phi = (\phi_1,....,\phi_n) : \mathbb{R}^m \to \mathbb{R}^n$. Suppose that $\phi_i$ is harmonic for each $i$, that is, $-\Delta \phi_i = 0$. Suppose we change from Cartesian to ...
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23 views

How to indicate the space of sesquilinear forms?

Is there a canonical way to indicate the space of sesquilinear forms? For bilinear forms I can use the (0,2) Tensor space, but since sesquilinear forms are not linear in one variable I don't know how ...
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Minimal surface with radially symmetrical function

The following image is from the book "Regularity Theory for Mean Curvature Flow", by Ecker. I consider the plateau problem, whose goal is to solve minimal surface given fixed boundary values. In ...
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1answer
27 views

How can I prove that for a Killing vector $\nabla^a \nabla_a \xi^\mu = -R^b_a \xi^a$?

I'm taking a course on General Relativity and I'm trying to prove that for a Killing vector field $\xi^\mu$ the following equation holds: $$\nabla^a \nabla_a \xi^\mu = -R^\mu_a \xi^a$$ Where ...
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1answer
44 views

Metric on real projective space

The standard metric on $RP^n$ is usually defined to be the metric that locally looks like the metric on $S^n$. But as a differentiable manifold (and not just as a set), $RP^n$ is not a subset of ...
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Riemannian metrics on homogeneous spaces

Let G be a Lie group and H be a compact subgroup. The (left) coset space G/H is, up to an isomorphism, equivalent to the smooth homogeneous manifold M. My question is, is it possible to impose an ...
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1answer
22 views

How to show the space of inverse-invariant metrics on a Lie group is infinite dimensional?

Let $G$ be a Lie group. I am trying to convinve myself there are 'many' Riemannian metrics on $G$ for which the inverse automorphism is an isometry. Denote the iverse by $i$. For any metric $g$ on ...
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32 views

Is exponential map an immersion?

Let $M$ be a connected Riemannian manifold. For $p\in M$, the injectivity radius at $p$ is the sup of the $\epsilon >0$ such that the Riemannian-distance ball $B_\epsilon (p)$ is a geodesic ball, ...
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The limit of a uniform convergent sequence of isometries is an isometry (problem 6-3 of Lee's “Riemannian manifolds”)

I'm trying to prove the following theorem: let $f_n : M \to N $ a sequence of isometries of Riemannian manifolds that converges uniformly to a function $f:M \to N$: prove that $f$ is an isometry too. ...
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1answer
30 views

Cotangent bundle tensor product tangent bundle

What is the meaning of Cotangent bundle tensor product tangent bundle: $T^*M\otimes TM$? what will an element of this space be?
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1answer
63 views

Levi Civita connection intuition and motivation

Can someone explain why need we the Levi-Civita connection and what it does intuitively?
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1answer
20 views

Can we always choose an isometric slice chart for a submanifold of $\mathbb{R}^n$

Let $S$ be a submanifold of $\mathbb{R}^n$. Let $p \in S$. Is there an isometric slice chart for $S$ in $\mathbb{R}^n$, around $p$? i.e, I am asking whether there is a diffeomorphism $\phi$ from some ...
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1answer
30 views

How to prove $(0,1) \times \mathbb{R} \, , \, (0,2) \times \mathbb{R}$ are not isometric?

I am trying to prove in an elementary way that $X_1=(0,1) \times \mathbb{R} \, , \, X_2=(0,2) \times \mathbb{R}$ (with the standrad euclidean metric inherited from $\mathbb{R}^2$) are not isometric as ...
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1answer
44 views

Riemaniann metric is an element of?

We are given the definition: A riemaniann metric $g$, is a map: $g:p\rightarrow<.,.>|_{T_pM}$ where $<.,.>|_{T_pM}$ is the usual bilinear symmetric etc.. It also says that the metric ...
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For which $a\in\mathbb{R}$ is does $D^*\subset\mathbb{R}^2$, with the Riemannian metric $\frac{du^2+dv^2}{(u^2+v^2)^a}$, have bounded distances?

$D^*\subset\mathbb{R}^2$ is the punctured unit disc, i.e. the set $ \{ (u,v) \in \mathbb{R}^2 | 0 < u^2 + v^2 < 1 \}$. Here's what I've got so far: two points in $D^*$ can be joined by a radial ...
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1answer
33 views

Normal coordinates and the metric matrix

While trying to follow and check the proof of Theorem 1 in this work on manifold averaging I reached the notion of normal coordinates. An important property is that the metric tensor at a point ...
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1answer
29 views

Approximation of piecewise smooth curves with same-lenght smooth curves in Riemannian manifolds

Let $M$ be a Riemannian manifold, and let $\gamma : [a,b]\to M $ be a piecewise smooth curve. Then, using Whitney's theorems, it can be proved that $\gamma$ is homotopic (by a homotopy relative to $a$ ...
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Diffusion on a Boundaryless Manifold and Tesselation

Suppose we are dealing with diffusion on a boundaryless manifold $M$. When people use the finite-element method to find an approximate solution, I always see them use \begin{align} \int_{M^*} W \Delta ...
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29 views

How to understand it will sweep out a 2-dim manifold?

As red line in picture below, I really can't imagine why it will sweep out a 2-dim submanifold. Below picture is from the 9th page of John M. Lee's 'Riemannian Manifolds: An Introduction to ...
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26 views

$X_z=\frac{d}{dt}_{|t=0} \Phi_t(z)$ has flow $\Phi_t$

Let $M$ be a manifold, $\Phi_t, t\in \mathbb R$ a one parameter group of diffeomorphisms and $X$ a vector field on $M$ definied by $$X_z:=\frac{d}{dt}_{|t=0} \Phi_t(z).$$ Show that $\Phi_t$ is the ...
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1answer
35 views

Geodesics are minimizing in a simply connected manifold without conjugate points?

Let $\tilde M$ be a compact Riemannian manifold, without conjugate points. Denote by $M$ its universal cover. Then in this paper, it is claimed that every geodesic is globally length minimizing. Why ...