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Questions tagged [riemannian-metric]

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2
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2answers
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Covariant derivative of determinant of the metric tensor

Let $(M,g)$ be a Riemannian manifold and $g$ the Riemannian metric in coordinates $g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$, where $x^{i}$ are local coordinates on $M$. Denote by $g^{\alpha \...
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1answer
19 views

Understanding a point of the example that the Euclidean metric is a Riemannian metric

A hint that many geometers give for people who start in Riemannian Geometry is associate the definitions of the course of Differential Geometry of curves and surfaces on $\mathbb{R}^3$ with the ...
3
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1answer
176 views

Computing the $2^\text{nd}$ fundamental form of $\mathbb{R}_+\times M^n\to \mathbb{R}^{n+p+1}$

Let $f: M^n\to \mathbb{S}^{n+p}$ be an isometric immersion. The cone over $f$ is defined to be the immersion \begin{align*} F:\mathbb{R}_+\times M &\to \mathbb{R}^{n+p+1}\\ (t,x)&\mapsto tf(...
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0answers
42 views

How is $dx^2+dy^2$ the Euclidean metric on $\mathbb R^2$

A Riemannian metric is a smooth symmetric covariant $2$-tensor field. If I put in two vectors, say $(1,2)$ and $(2,1)$, I don't get $\|(1,2)-(2,1)\|=\sqrt{2}$: $$(dx^2+dy^2)((1,2), (2,1))=2+2=4.$...
2
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1answer
68 views

Integration by parts for tensor fields on Riemannian manifold

I'm working on the following exercise in my Riemannian manifolds book: Suppose $M$ is a compact, oriented Riemannian manifold with boundary. Show that if $\omega$ is any $k$-tensor field and $\eta$...
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1answer
36 views

Normal Derivative of Scalar Curvature at a Minimal Surface

Consider a Riemannian 3-manifold $M$ with boundary $\partial M$ with unit normal $n$, and mean curvature $H$. Consider the scalar curvature $R$. I'm interested in computing $n^a \nabla_a R = \langle ...
2
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2answers
68 views

Computing volume of Riemannian manifolds

I have a question- It is given that $f: M \mapsto N$ is an $n$- sheeted covering map and a local isometry then I have to show that volume$(M) = n$ volume$(N)$, where $M$ and $N$ are Riemannian ...
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1answer
24 views

Surface covered by hyperbolic plane admits a hyperbolic metric

Let $S$ be a surface. Is it true that if $S$ is covered by the hyperbolic plane (or a subset thereof) then it admits a Riemannian metric of constant negative curvature? How does the metric (or ...
4
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1answer
66 views

Intuition behind Riemannian-metric

I apologise in advance if something like this has been asked already and I will delete this question immediately if an already answered question of this sort clears my doubt, which is- What is a ...
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0answers
32 views

How to find out this Riemannian metric for this manifold?

Came across an example of Riemannian metric example today, one of them was of a Riemannian metric defined on upper half plane $ \mathbb{H^{2}} = \{ (x,y): y>0\} $ as: $$ ds^2 = \frac{dx^2 + dy^2}...
2
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1answer
60 views

Negative Curvature on Torus

We were looking for an example of a compact non-simply connected riemannian manifold with non positive sectional curvature. We came up with the following idea, which is wrong by Gauss-Bonnet, but we ...
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2answers
46 views

Gaussian curvature of a given metric

The exercise is from do Carmo, Differential Forms and Application, p.97. Consider $\mathbb{R}^2$ with the following inner product: if $p=(x,y)\in\mathbb{R}^2$ and $u,v\in T_p\mathbb{R}^2$, then $$\...
2
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1answer
48 views

The riemannian metric of a neighborhood of the boundary of a compact manifold

Let $M$ be a compact riemannian manifold with boundary $\partial M$. We have that $\partial M$ is also compact and I was able to show that there is some $a>0$ such that the map $F:[0,a]\times \...
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0answers
33 views

If I shift a curve in the normal direction, are its tangent vectors also shifted by parallel transport?

Deleting the context, I'm in the following situation: $\Sigma^{n-1}$ is a hypersurface of a riemannian manifold $M^n$ and $\nu:\Sigma\to T\Sigma$ is a vector field normal to $\Sigma$ with $|\nu(p)|=r&...
3
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1answer
51 views

Existence of boundary cylindrical neighborhood for a compact manifold

Let $M$ be a compact riemannian manifold with boundary $\partial M\neq \varnothing$. I would like to show that there is some neighborhood $U$ of $\partial M$ which is diffeomorphic to $[0,a)\times \...
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2answers
114 views

Intuition behind riemannian metric

I formally understand what a riemannian metric $g$ on a manifold $M$ is. It's basically a section of the vectorbundle $T^*M\otimes T^*M \to M$ (which in the end corresponds to mapping 2 vectors from a ...
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1answer
25 views

Confusion regarding summation convention.

The question goes as follows: If $\theta$ is the angle between two non-null vectors $A^i$ and $B^i$, show that $sin^2 \theta = \frac{(g_{ij}g_{kl}-g_{ik}g_{jl})A^iB^kA^jB^l}{(g_{ij}A^iA^j)(g_{kl}B^...
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0answers
259 views

The Taylor expansion of a metric: a natural viewpoint?

A Riemannian metric $g$, in normal coordinates, has a Taylor expansion $$g_{ij}(x) \ = \ \delta_{ij} \ - \ \frac{1}{3}R_{iklj}x^kx^l \ - \ \frac{1}{3}R_{iklj;m}x^kx^lx^m \ + \ \frac{1}{180}\left(8R_{...
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0answers
56 views

Construction G-Invariant Riemannian Metric

Let $M$ be a smooth manifold and $G$ be a lie group acting transitively on $M$. I know by Corollary 1.27 of these notes that there to exist a Riemannian metric $g_G$ on $M$ satisfying the in-variance ...
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0answers
40 views

Flat tori as a riemannian product

A flat torus is defined as the torus with the metric inherited from its representation as the quotient $\mathbb{R}^2/\Lambda$ where $\Lambda$ is a discrete subgroup of $\mathbb{R}^2$ which is ...
0
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1answer
141 views

Metric on unit circle

How do you define a Riemannian metric for the unit circle. Is it $ds^2=dx^2+d\theta^2$? I want to also measure the length of the vector from the origin. This would be a standard euclidean metric ...
2
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0answers
89 views

Sign of the Bakry-Emery curvature operator $\Gamma_2(f) :=|Hess f|^2+Ric(\nabla f,\nabla f)$

I am not an expert of Riemannian geometry (coming mainly from functional analysis in $\mathbb R^n$). Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with Ricci curvature bounded from ...
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0answers
35 views

Induced metric from Euclidean space

Suppose $F:(M^{n-1},g)\rightarrow R^n$ is an embedding, $g$ is the metric induced by $F$ from $R^n$, I want to show that $$g(\nabla_{\partial_i}\partial_j,\partial_k)=<\partial_i\partial_jF,\...
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0answers
33 views

The distance between 2 points with non zero curvature

Suppose we have 2 points which they are connected by an curve. So it’s not line deferment! How I can find the distance between these 2 points? Is there any general formula? Suppose this problem for ...
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0answers
85 views

Derivative of the Exponential map explained

I'm trying to understand how to interpret the derivative of the exponential map in terms of the Jacobi field? Define the map $$ f(x):Exp_p(x), $$ where the Riemannian manifold is on $\mathbb{R}^d$ ...
1
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1answer
62 views

Derivative of boundary volume element

If $(M,g)$ is a 2-dimensional surface with boundary, now we have a deformation of metric $(g_t)$, suppose $\frac{dg_t}{dt}\big|_{t=0}=h$ which is a symmetric $(0,2)$ tensor, then the derivative of ...
1
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1answer
63 views

Computations with respect to a Riemannian metric

I have the following Riemannian metric on $\mathbb{R}^2$ : $$g = 4\frac{1}{(1+r^2)^2}(dx^2+dy^2),\text{ with }r^2 = x^2 + y^2.$$ At every point of $\mathbb{R}^2\backslash \{(0,0)\}$, let $$e_r = \frac{...
1
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1answer
247 views

Proving that a map is an isometry.

Let $D$ be the open unit disk in the complex plane and let $f: D \rightarrow \mathbb{C}, z \mapsto \frac{i+z}{1+iz}$. We equip the upper half plane $H = \{z \in \mathbb{C} \mid \operatorname{Im}(z) &...
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1answer
66 views

quadratic form inequality for SPD matrices

Let $x = (0,0,...,0,1)^T$ and $S,R$ symmetric positive definite matrices. I want to see if the following is true: $\frac{1}{2} x^ T\left(R + S \right)x \geq x^T \left(S^{\frac{1}{2}} \left(S^{-\frac{...
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0answers
49 views

Proving that a parallel vector field has constant length

I am reading through McInerney's First steps in Differential Geometry. Proposition 5.4.4 states that in a Riemannian Space $(U,g)$ with an adapted connection $\nabla$, a parallel vector field $V$ to a ...
1
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1answer
153 views

Local representation of a riemannian metric (change of coordinates)

Let's say I have a manifold $M$ and an atlas $\mathcal{A} = \{ (U,\varphi), (V, \phi)\}$ of $M$. I am asked to show that there is a unique riemannian metric $g$ such that its representations in the ...
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2answers
289 views

Is orientability needed to define volumes on riemannian manifolds?

In the book Riemannian Geometry, by Mandredo do Carmo, he supposes that $M$ is a riemannian oriented manifold and then defines the volume of a region $R$ contained in some image $\boldsymbol x(U)$ of ...
2
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1answer
278 views

Curvature and Christoffel symbols

Let (M,g) be a Riemannian manifold and $\nabla$ its Levi-Civita connection. Suppose that the curvature $R$ is zero. Is $\nabla$ locally trivial?, i.e., do the Christoffel symbols equal to zero? Thank ...
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0answers
76 views

Riemannian-connection equations with one-forms (basic)

I am learning the basics of reformulating vector field equations in terms of differential forms. In particular, I am trying to express the equations for the Riemannian connection using differential ...
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1answer
91 views

Decomposition of the tangent space of a manifold with inner product

I'm reading about isometric immersions in the Do Carmo's Riemannian Geometry and there he states that if $f: M^n \longrightarrow \overline{M}^{n+m=k}$ is an immersion and $\overline{M}$ endowed with ...
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1answer
67 views

Riemannian manifolds and the tangent vector of a differentiable curve

Let $φ$ be a differentiable curve on a riemannian manifold $M$ defined on an open interval $]a,b[$ , and let $v_t$ be the tangent vector of $φ$ at $φ(t)$. Let $B$ be the basis of $T_p(M)$ How can ...
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0answers
132 views

Pseudo Riemann metric and Riemann metric

I have two questions The first question : Who can give me an example of a manifold is not paracompact and we can't define a Riemann metric on it ? The second one I want read about pseudo ...
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0answers
91 views

Could is a riemannian metric uniquely determined by its exponential map?

It is very known fact that isometric metrics have the same exponential maps. I am interesting in the converse, if we know the exponential map could we recover the metric? Suppose that $(M,g_1)$ and $(...
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0answers
150 views

Geodesics of manifold and geodesics of its submanifold with induced metric

Suppose we have a riemmanian manifold $(\mathcal{M}, g)$. Let $\mathcal{N}$ be properly embedded submanifold of $\mathcal{M}$ with induced metric $\bar{g}$ from $g$. ($\mathcal{N}$ is not ...
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2answers
3k views

Definition of gradient of a function $f$ in Riemannian manifold

I'm reading Semi Riemannian Geometry with applications to relativity by Barret Oneill and I'm trying understand the definition of gradient of a function $f$ in Riemannian Manifold. I know that ...
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1answer
237 views

A formula of Riemannian metric and its determinant.

Let $g_{ij}$ be a Riemannian metric of a Riemannian manifold, prove: $\large 
g^{ij}\frac{\partial g_{ij}}{\partial x^k}=\frac{1}{G}\frac{\partial G}{\partial x^k},
G=det(g_{ij}), g^{ij}=g_{ij}^{-1}
$ ...
1
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1answer
55 views

Is every function Lipschitz in some Riemannian metric?

Let $M =\mathbb{R}^{n}$ and let $f : M \to \mathbb{R}$ be a smooth function. Define the Riemannian metric $$ \|\delta x\|_{x} = (1+ \|\tfrac{\partial f}{\partial x}(x)\|)\|\delta x\| $$ on $M$ (here ...
3
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0answers
386 views

Is a metric tensor a norm of the tangent space?

In a (pseudo-) Riemannian manifold, we have a metric tensor defined on each point in the manifold. I.e. a tensor field. With this metric tensor we can define also a metric on the manifold by ...
3
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1answer
75 views

Why is an interval invariant on coordinate transformations?

I learned that in (semi-)Riemannian geometry an interval $ds^{2}=g_{ab}dx^{a}dx^{b}$ is invariant on coordinate transformations. But there is something I do not get in the concept. Consider the simple ...