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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Metric on Heisenberg group arising from CR isomorphism to punctured odd-dimensional sphere

It is well known that odd-dimensional spheres $S^{2n+1} \subset \mathbb{C}^{n+1}$ are CR manifolds, and on removing a point we get a CR isomorphism to the (underlying manifold of the) Heisenberg group ...
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Clarification about the definiton of a metric on a vector bundle

Let $E$ be a vector bundle over a manifold $M$. By definition, a metric on $E$ is a function $g:E \times _M E \to M \times \mathbb{R}$. In wikipedia, they say $g$ is a bundle map. However, it is not ...
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The arc length in Riemannian geometry is well defined (independent from the choice of the coordinates)

Let be $(M,g)$ a connected Riemannian manifold and $p,q \in M$. 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 ...
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Isomorphisms of bundles

Studying on vector bundles came across the following problem. Question: Let $M_1$ and $M_2$ Riemannian manifolds with $TM_1^{\perp}$ and $TM_2^{\perp}$ their normal of ranks $k_{1}$ and $k_{2}$, ...
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20 views

Metric compatibility of dual connection

Let $(M,g)$ be a Riemannian manifold with Levi Civita connection $\nabla$. Then $\nabla$ satisfies a compatibility condition: $(\nabla_ZX,Y)+(X,\nabla_ZY)=Z((X,Y))$ where $(\cdot,-)$ is a Hermitian ...
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An compute of Riemannian geometry [on hold]

According to Einstein summation convention , $g_{ij}$ is metric tensor,and $f$ is a real function. Show that : $$ g_{jl}g^{ij}\frac{\partial f}{\partial x^i}g^{kl}\frac{\partial f}{\partial ...
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1answer
20 views

Laplace-Beltrami operator as sum of orthogonal projections

Let $M$ be a submanifold of $\mathbb R^l$ with the induced metric. Let $(\xi_\alpha)$ be the standard orthonormal basis on $\mathbb R^l$. For each $x \in M$, let $P_\alpha(x)$ the projection of ...
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1answer
22 views

Verification of some conditions and facts of the Laplacian on a Riemannian manifold

I came across the following introduction to a paper I was reading: "Let $L$ be a second-order, elliptic, differential operator, with smooth coefficients, on a Riemannian manifold $M$. Assume $-L$ is ...
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3answers
128 views

uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
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Uniqueness of covariant derivative in Do Carmo

2.2 Proposition: Let $M$ be a differentiable manifold with an affine connection $\nabla$. There exists a unique correspondence which associates to a vector field $V$ along the differentiable curve ...
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27 views

Weierstrass transform on a Riemannian manifold

As it has been written on this Wiki article, the Weierstrass transform can be defined on a Riemannian manifold. Even though, I couldn't find any references I guess this transform for a smooth function ...
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1answer
40 views

Metric tensor for n-sphere in ambient coordinates

Let $S^n$ be the unit n-sphere embedded in $\mathbb{R}^{n+1}$: $$ S^n = \{ a \in \mathbb{R}^{n+1} \mid a \cdot a = 1 \} $$ What is the induced metric tensor for the sphere, in $\mathbb{R}^{n+1}$ ...
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20 views

Mean curvature formula of hypersurface in sphere

I wonder if there is a general bound for mean curvature of a hypersurface embedded in a sphere. Could anyone give me a reference?
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1answer
30 views

Concavity of distance function in $\mathbb{R}^n$ or determinant of $(x^T \cdot x)$

I would like to compute the concavity of the distance function in $\mathbb{R}^n$. Let $ f(x) =- \Vert x \Vert $ in $\mathbb{R}^n$. Then $\nabla_xf=- \frac{x}{\Vert x \Vert}$. And ...
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1answer
55 views

Is there a natural Riemannian structure on the total space of a vector bundle?

Suppose $B$ is a Riemannian manifold and $\pi: E \to B$ is a smooth vector bundle equipped with a metric. Is there a natural Riemannian metric on $E$, i.e. a bundle metric on $TE\to E$? It seems ...
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9 views

Reference of integral on differential manifolds and conformal aplications

I need goods and fast reference about integral of differential manifolds, more precisely about results of change variable but not with differential forms. I need goods and fast reference about ...
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1answer
24 views

Computation of Liebracket for Vectorfields assosiated with a Variation of Geodesics

Let $(M,g)$ be a Riemannian manifold, $V \subset \mathbb{R}^2$ be an open subset and $\alpha: V \rightarrow M; (s,t) \mapsto \alpha(s,t)$ a smooth map. for $(s,t) \in V$ one can define $$ ...
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1answer
24 views

Volume density on a Riemannian manifold as a measure

I am having some trouble in seeing exactly how the Riemannian density form gives rise to a measure on $\text{Borel(M)}$. Let $(M,g)$ be a Riemannian manifold. We have the Riemannian density $\mu_g$. ...
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1answer
22 views

points which are fixed points of a finite group action

consider an open set $\tilde{U}\subset\mathbb{R}^n$ and a finite Lie-group $G$, which acts smoothly on $\tilde{U}$, i.e. we have a smooth map $G\times \tilde{U}\rightarrow\tilde{U}$. Suppose further, ...
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18 views

Diffeomorphism and Orientable double cover

Suppose that the orientable double cover of two homeomorphic surfaces are diffeomorphic, is it true that these surfaces are diffeomorphic?
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17 views

One-sided surfaces and the second variation area formula.

I know how to find the second variation area formula for a two-sided minimal embedded surface in a 3-manifold and the condition for such a surface to be stable. But, what about one-sided surfaces? ...
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Motivations for Hyperbolic Geometry

Why would one study hyperbolic geometry? I am only aware of the motivation where you give axioms for elementary euclidean geometry and then start to wonder wether the parallel axiom is necessary. You ...
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Geodesics of Sasaki metric

I would like to ask the community for a reference on the following question: Let $(M,g)$ be a Riemannian manifold and $(T^1M,g_S)$ be the unit tangent bundle with the Sasaki metric. Is it true that ...
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31 views

Geodesics without a metric

By definition, a geodesic is a mapping $\gamma: I = (0, 1) \rightarrow M$ such that $\nabla_{\dot{\gamma}(t)} \dot{\gamma}(t) = 0$. Here we only need the connection. So, we do not need a metric to ...
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Exponential map on a sphere in spherical coordinates

Let $M = \{ (x^\varphi, x^\theta) : x^\varphi \in [0, \pi), \thinspace x^\theta \in [0, 2\pi) \}$ be a manifold with metric $\mathrm{d}s^2 = (\sin x^\varphi)^2 (\mathrm{d} {x^\theta})^2 + (\mathrm{d} ...
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1answer
27 views

Reference on manifolds with boundary

I am here because I want to know if someone knows of some good e fast books or references about manifolds with boundary. Help me please.
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3answers
67 views

Can we bypass connection?

I am new to differential geometry. It is surprising to find that the linear connection is not a tensor, namely, not coordinate-independent. Can we bypass this ugly object? Only intrinsic quantities ...
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2answers
135 views

What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are ...
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Advanced Differential Geometry Textbook

In algebraic topology there are two canonical "advanced" textbooks that go quite far beyond the usual graduate courses. They are Switzer Algebraic Topology: Homology and Homotopy and Whitehead ...
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1answer
28 views

Is such kind of manifold Riemannian? Deforming the metric on the unit square by a weight applied in one direction

If the metric is defined on a bounded subset of the x-y plane,let's say a closed square area $0\le x,y\le1 $, the metric is defined as $$\langle u,v\rangle =\langle (u_x,u_y),(v_x,v_y)\rangle =\langle ...
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13 views

Solutions to Dirichlet problem on manifolds with boundary

I am looking for a reference for the following assertion: Let $M$ be a Riemannian manifold with boundary, and $f:\partial M \rightarrow \mathbb{R}$ be smooth. Then there exists a unique smooth ...
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Parallel transform of a vector by Lie derivative

I am new to differential geometry and I learn by myself. It seems that we need something extra called a connection to parallel transport a vector along a curve. But, suppose we have a vector field ...
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53 views

About the diameter of a Riemannian manifold

My exact problem is a Riemannian manifold defined on $SU(2^n)$, where the metric is defined as follows: If $U(t)$ is a curve so that $U′(t)=−iH(t)U(t)$ (so just a unitary evolution of a quantum system ...
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1answer
72 views

flat manifold, curvature and the circle

A Riemannian manifold is said to be flat if the curvature is 0 everywhere. An example in dimension 1 is the circle. However, I cannot see how the curvature of the circle could be 0. See for instance ...
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Surfaces obtained by $\gamma$-reduction

$\mathcal{C}$ will denote the collection of all connected compact (not necessarily orientable) smooth 2-dimensional surfaces-without-boundary embedded in $M$ ( here $M$ is a complete Riemannian ...
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338 views

The Riemann Sphere Interpretation

Is the Riemann sphere anything more than a simple visual tool to help students understand the complex planes, or the behavior of complex valued functions at infinity, limit points etc? Or is there a ...
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1answer
38 views

Equivalent conditions for a linear connection $\nabla$ to be compatible with Riemannian metric $g$

I am reading John M. Lee's Riemannian Manifolds: An Introduction to Curvature. In Lemma $5.2$, it is said that the following conditions are equivalent for a linear connection $\nabla$ on a Riemannian ...
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32 views

Singularity in Ricci flow vs Ricci soliton

In the paper "The formation of singularity in Ricci flow" Hamilton studied systematically the possible singularities of the flow.My question is why it is important to classify Ricci solitons in order ...
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2answers
33 views

contraction of the Riemann-Christoffel tensor

I'm attempting to prove that a particular contraction of the Riemann-Christoffel tensor is zero. I know that when the top and second of the bottom indices are contracted we get the Ricci tensor. But ...
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1answer
52 views

A question on Stokes theorem for Lipschitz functions

Let $M$ be an oriented compact Riemannian manifold. Let $f$ be a Lipschitz function on $M$, denote $M'\subset M$ be the set on which $f$ is differentiable. On one hand, Stokes theorem works for ...
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23 views

Confusion about the total mean curvature of spheres in a manifold

I am browsing through a paper and I am confused by a notation the meaning of which I do not understand. Let $S_{p,\rho}$ be the geodesic sphere of center $p$ and radius $\rho$ in a Riemannian ...
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Proof of $d^\ast A =0$ where $D=d+A$ is Yang Mill connection

Recall $$ F =( dA_{ij} + A_{il}\wedge A_{lj} )\mu_i \otimes \mu_j^\ast $$ Hence if rank of $E$ is $2$, then $$ F= dA $$ since $A$ is skewsymmetric. If $D$ is Yang Mill connection then $ D^\ast F=0$. ...
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Ellipticity of an operator in Gunther's proof of the isometric embedding

In Deane Yang's notes about Gunther's proof of the celebrated isometric embedding theorem, at the end it is stated that $v$ inherits the regularity of $h$ because the operator $I-Q_0(v,\cdot)$ is ...
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29 views

The gradient in $n$-dimensional spherical coordinates

I am in the middle of a computation where I need to work with the formula of the gradient in spherical coordinates in $\Bbb R ^n$ (no preferred convention for the angles). I could patiently and ...
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1answer
28 views

Proof of the fundamental inequality of the index form

I am looking for a proof of the fundamental inequality of the index form, which I have seen as references or statements in a lot of sources, but without a proof. This is the statement: Let $M$ be a ...
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1answer
30 views

saddle point geometry of $f(x,y)$ when partial derivatives are of same sign.

The sign of Hessian matrix and sign of any partial derivative of function $f(x,y)$ gives the informations about maxima, minima, saddle points of function and sometimes perplexed informations when it ...
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113 views

Index of zero of a gradient vector field at a critical point

Let $M$ be a Riemannian manifold with a Morse function $f: M \to \mathbb{R}$. The zeroes of the gradient vector field of $f$ are the critical points of $f$. How do you show that a critical point ...
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1answer
133 views

Newton iteration on Riemannian manifolds

Suppose $f:M \to N$ is a smooth map between complete Riemannian manifolds of the same dimension. Suppose $Df(m_0)$ is invertible, and $n$ is a point close to $f(m_0)$. Can we perform Newton iteration ...
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1answer
43 views

ricci tensor of 2-sphere $S^2$

Hi could someone show me explicitly how to compute the ricci tensor $g_{ij}$?
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2answers
62 views

Mean Curvature flow: Evolution equation of any invariant symmetric homogeneous polynomial with input the Weingarten map.

I have the following evolution equations realted to mean curavture flow, with the induced metric $g=\{g_{ij}\}$, measure $d\mu$ and second fundamental form $A=\{h_{ij}\}$: 1)$\frac{\partial}{\partial ...