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 do conformal maps affect curvature?

Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemanniam connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with ...
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1answer
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Local isometry between simply connected manifolds

Suppose $D:\tilde{M}\rightarrow N$ is a local diffeomorphism between two simply connected smooth manifolds $\tilde{M}$ and $\tilde{N}$. $D$ is onto. In the case of $D$ being a covering map, it follows ...
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Intuitive understanding into the mean curvature flow

I am trying to develop some intuition into the properties of mean curvature flow of a surface in $\mathbb{R}^3$. As an example, I am trying to understand what happens to a surface of revolution $$S = ...
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2answers
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The configuration space of directed great circles on the sphere

I am reading a proof of Pu's inequality in Katz' book "Systolic geometry and topology". In section 6.4, he describes a fibration $q : SO(3) \to S^2$ which I can't quite figure out. First, he thinks ...
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How to define and compute the norm of a vector with riemannian metric?

Let us consider for example, the riemannian metric $g=e^xdx^2+dy^2$ (it is symmetric and definite positive), with associated matrix $\begin{pmatrix} e^x & 0\\ 0 & 1 \end{pmatrix}$. Consider ...
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Mean and Gaussian curvature - normalization to interval $[0, 1]$ [on hold]

I can compute curvature of the $2.5D$ surface. Problem is, I need the results scaled in interval $[0,1]$ (or $[-1,1]$). Is it possible to compute this directly or I need to compute all curvatures of ...
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1answer
26 views

The Levi-Civita Connection on the Hyperbolic Plane

In this question here, I asked about computing the Levi-Civita connection matrix on the Hyperbolic Plane, defined as $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = ...
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1answer
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Linear Connection on the Hyperbolic Plane

For the upper half-plane $\mathbb{H}^2=\{x+iy\in\mathbb{C}\ \vert\ y>0\}$ equipped with the metric $g = \frac{1}{y^2}(dx^2+dy^2)$, I computed the Christoffel symbols as follows: ...
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Warped product of a manifold with flowing surfaces

We consider a Riemannian 3-manifold and some 2-surfaces $N_t$ embedded in the manifold that flow with time $t \geq 0$. If $v$ is the normal speed with which the surfaces flow, and we assume that this ...
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1answer
15 views

Schur's theorem in DoCarmo's “Riemannian Geometry”

The exercise 8 of chapter 4 of Do Carmo's "Riemannian Geometry" ask to prove the Schur's Theorem. I don't understand a step in the hint (the "hint" is essentially the proof of the theorem). Schur's ...
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23 views

Sectional Curvature, Gauss curvature

I have a problem with a computation which shows that the sectional curvature coincide with the Gauss Curvature in dimension 2. This is the definition of sectional curvature I am using: ...
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Examples of global properties that don't arise from local knowledge

Let $M$ be a smooth manifold. As an example of a global property that arises from local data we know that if $(M,g)$ is a compact surface without boundary then the Euler characteristic is given by $$ ...
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Exercise 5, chapter 4 in Do Carmo's “Riemannian Geometry”

This is probably a silly question and maybe in the end the answer is trivial but I can't see it. The problem is the following. Let $M$ be a Riemannian manifold, $\gamma \colon[0,l]\to M$ be a ...
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27 views

Harmonic maps between Riemann surfaces

In 'Compact Riemann surfaces' Jost defines harmonic maps between surfaces $S_1,S_2$, with local coordinates z on $S_1$ and metric $\rho^2|du\,d\overline{u}|$ on $S_2$ as $u\in C^2$ solving the ...
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2answers
37 views

How to understand conjugate points on a Riemannian manifold?

I'm having trouble grasping what it means for two points to be conjugate on a Riemannian manifold. Could someone provide a geometric or intuitive explanation for this? For clarification: given a ...
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1answer
21 views

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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17 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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20 views

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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1answer
32 views

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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1answer
19 views

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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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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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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41 views

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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1answer
59 views

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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31 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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2answers
32 views

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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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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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 ...
3
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1answer
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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
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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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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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1answer
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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
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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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1answer
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
23 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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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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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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1answer
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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
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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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2answers
37 views

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
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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 ...