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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Is it mathematically correct to say that if the metric is flat/curved the *shortest* path is/not a Euclidean straight line?

Is it mathematically correct to say that if the metric is flat/curved the shortest path is/not a Euclidean straight line? I am still hesitant to make this claim, due to at least one counter example. ...
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0answers
53 views

Covariant derivative and box operator commutator

I know that the commutator of two covariant derivatives is giving some Riemann tensors as follow: ...
3
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2answers
63 views

Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = ...
3
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1answer
37 views

Mean Curvature Flow

Recently I am reading the mean curvature flow from the lecture notes of Carlo Mantegazza where I found that Under mean curvature flow given by$$\begin{cases}{\partial\over \partial ...
0
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1answer
32 views

(Co)Tangent bundle of Cone manifold

Given a Riemannian manifold $(M,\bar{g})$, we can construct the Riemannian cone manifold $(C(M), g )$ as follows. Topologically, $C(M)$ is $M \times \mathbb{R}_{>0}$. We equip this with the ...
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1answer
20 views

Show that the section $g(x_1,x_2,x_3)=x_1^2dx_1^2+dx_2^2+dx_3^2$ defines a Riemannian metric on $\mathbb{R}^3 - \{x_1=0\}$

Show that the section $g$ of $T^*\mathbb{R}^3 \otimes T^*\mathbb{R}^3$ defined by $g(x_1,x_2,x_3)=x_1^2dx_1^2+dx_2^2+dx_3^2$ defines a Riemannian metric on $\mathbb{R}^3 - \{x_1=0\}$ and compute ...
2
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0answers
29 views

use of existence of bi-invariant differential form on a Lie group?

In do-carmo's Book "Riemannian Geometry" there is an exercise on proving existence of a bi-invariant metric on any compact connected Lie group. (pg 46, question 7). In the first stage, you are ...
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27 views

Uniqueness of bi-invariant metrics on Lie groups?

As noted here , a Lie group $G$ admits a bi-invariant metric if and only if $G$ is the cartesian product of a compact (Lie) group and a vector space $\mathbb{R}^n$. The question: For which Lie ...
3
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2answers
43 views

Notion of curvature for a volume embedded in $R^3$

This question might sound slightly vague, but please bear with me. If I have an orientable, closed, sufficiently smooth surface in $R^3$, I can define its principal curvatures, mean curvature as ...
4
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1answer
28 views

Calculate the length of $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ with the metric $g=\frac{dx^2+dy^2}{y^2}$ and compare with euclidean metric

Consider the metric $g=\frac{dx^2+dy^2}{y^2}$ on $\mathbb{R}_+^2=\{(x,y) \in \mathbb{R}^2 : y>0\}$. Calculate the length of the curve $\gamma(t)=(t,t), t \in [-1,-\frac{1}{2}]$ and compare ...
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1answer
62 views

Curvature and Circumference of Circle

Theorem Let $\gamma\colon [a,b]\rightarrow \mathbb{R}^2$ be a unit speed simple closed curve, with $\gamma'(a)=\gamma'(b)$ and $N$ is the inward-pointing normal. Then $$ \int_{a}^b ...
2
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1answer
27 views

An inequality for absolute total curvature in Riemannian surfaces

Let be $M\subseteq \mathbb{R}^3$ a compact (Riemannian) surface and let be $K$ the gaussian curvature of $M$. I want to prove that $$ \int_{M} |K| \geq 4\pi(1+g(M))$$ where $g(M)$ is the genus of ...
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1answer
66 views

Ricci tensor and average of a tensor

Let $(M^n,g)$ be an oriented Riemannian $n$- manifold and $g$ is a Riemannian metric on $M$ , $\mathrm{d}\sigma$ is Riemannian volume form on $S^{n-1}$ and $\text{Vol}(S^{n-1})$ is volume of ...
2
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0answers
30 views

a problem on geometry of hypersurfaces

Recently I am reading book on mean curvature flow by carlo mantegazza.There I found a problem on hypersurfaces stated below : Show that if the hypersurface $M \subset {R}^{n+1}$ is locally the graph ...
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1answer
38 views

Isometry in Hyperbolic space

Let $\mathbb{H}^2=\{ (x,y)\in\mathbb{R}|\ y>0 \}$ the hyperbolic space with the metric $g=(dx^2+dy^2)/y^2$. Let ...
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0answers
21 views

Two successive isometric immersions: relation between mean curvature vectors?

Let $M_0$ be a Riemannian manifold, $M_1$ a geodesic sphere of $M_0$ and $M_2$ an isometrically immersed submanifold of $M_1$, ie: $$ M_2 \subset M_1 \subset M_0$$ Take $X \in M_2$, and: $T_2$ the ...
10
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135 views

On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
0
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0answers
35 views

Generalizing Pearson's coefficient to determine properties of embedded manifold

I have the following dilemma: We know that for random vectors we have Pearson's coefficient of skewness. I think you all agree that in some sense it measures the shape properties of the ...
6
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0answers
70 views

Vectors that geodesically generate the same surface

Suppose that $\langle M,g \rangle$ is a complete, simply connected Riemannian symmetric space. The surface geodesically generated by a vector $\xi$ in $T_pM$ is the set of points lying on geodesics ...
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27 views

Space of $G$-invariant Riemannian metrics contractible?

A well-known result in (psuedo)Riemannian geometry is that the moduli space of (pseudo)Riemannian metrics on a smooth manifold is contractible. In the case when you have a smooth action of a group $G$ ...
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1answer
37 views

Formal name for the coordinate values of the pushforward of the inverse metric on an embedded manifold?

What is the formal name of the following object: \begin{align}\tag{4} \Delta^{\alpha \beta} = \dfrac{\partial y^\alpha}{\partial x^m} g^{mn} \dfrac{\partial y^\beta}{\partial x^n} \end{align} where ...
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0answers
24 views

How to prove Bochner and Laplace De-Rahm coincide on functions

I am having trouble seeing why Bochner Laplacian and Laplace De-Rahm operator coincide on functions. De-Rahm operator is defined via hodge dual and exterior derivative: ...
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1answer
48 views

How many ellipsoids can be maximally inside a circle?

This discussion is related to this discussion here where I want to deduce the area difference between such two circles filled with ellipsoids. Actually, to understand this difference is the main ...
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0answers
44 views

Taylor expansion of Riemannian exponential map and Jacobi fields?

1) Let $(M,g)$ be a Riemannian manifold, and at $p\in M,$ let $exp_p:U\subset T_pM\to M$ denote the Riemannian exponential map defined on $U\subset T_pM.$ I'd like to know how I could expand $t\mapsto ...
2
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1answer
31 views

Covariant derivative of a constant inner product and how it decomposes into local coordinates

I wanted to confirm that my explicit (symbolic) computations of the covariant derivative of a constant inner product is (carefully) done right and correct. For the physicists (includes me), this is ...
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0answers
31 views

Showing that the rank of the complex projective space is 1

I was assigned the task of calculating the rank of the complex projective space $\mathbb C P^n=SU(n)/S(U(1)\times U(n-1))$ and am not sure how best to approach that task. (looking in the ...
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1answer
89 views

The Weierstrass-Enneper representation, the Gauss map

Lemma: Let $x:S\to\mathbb{R}^3$ be a conformal minimal immersion of a Riemann surface. The 1-forms $f_k=(x_{k,u}-ix_{k,v})dz$ satisfy: $$ \sum_kf_k^2=0\qquad (1)\qquad \&\qquad ...
0
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0answers
33 views

Normal coordinates

I was wondering if this is a legitimate way to define the induced basis of the tangent space in normal coordinates. So the exponential map is a diffemorphism $exp:U \subset T_pM \rightarrow V \subset ...
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0answers
31 views

Relation between parallel transport and Jacobi field II

Before I asked a question here: Relation between parallel vector field along a geodesic and Jacobi field along that same geodesic The current question is related, and actually arise from numerical ...
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36 views

Homogeneous metric on a homogeneous space $G/K$ - is this the same as a $G$ - invariant metric?

I have trouble putting down the notion of a homogeneous Riemannian metric. Suppose we are given a Riemannian manifold $(M,g)$ on which a compact Lie group $G$ acts transitively by isometries (this ...
16
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1answer
114 views

Are there spaces that 'look the same' at every point, but are not homogeneous?

A metric space is homogeneous if for any two points there is a global isometry that maps one into the other. It is locally homogeneous if any two points have isometric neighborhoods, i.e. the space ...
5
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1answer
70 views

Can a $1d$ space never be curved?

I was wondering about this: Wikipedia article I refer to (here I refer to the first part: metric) This wikipedia article claims that this hyperbolic space model has constant curvature $-1.$ I believe ...
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0answers
43 views

Does a homogeneous metrizable space admit a compatible homogeneous metric?

Assume that X is a compact metrizable topological space for which the action of homeomorphism group is transitive. Is there a compatible metric d on X such that the action of group of isometries ...
1
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1answer
49 views

Jacobi field along every geodesic?

I stumbled over the question: Given a manifold $M$. Can there exist a vector field that is a Jacobi field along every geodesic? Now, the answer is apparently: Yes, because the $0$ vector field does ...
1
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1answer
38 views

Normal coordinates and the metric tensor

I was wondering whether the metric tensor in normal coordinates can be expressed somehow in terms of the exponential map. Cause I just don't see how the metric in normal coordinates is actually ...
1
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0answers
30 views

Isometry algebra implication from Riemannian covering

I really wish that, if $\pi:(M,\mathrm{g})\twoheadrightarrow(N,\mathrm{h})$ is a Riemannian covering, then $\mathfrak{i}(N,\mathrm{h})\leq\mathfrak{i}(M,\mathrm{g})$, where ...
1
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1answer
22 views

Relationship between euclidean metric in sphere of radius $r$ and the unit sphere.

I want to show $g_r=r^2g_1$ where $g_1$ is the (Riemannian) metric in the unit sphere induced by its inclusion in $\mathbb{R}^n$ and $g_r$ is the metric in the sphere of radius $r$ also induced by ...
0
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1answer
25 views

Gradient of Distant Function

I am learning the Hessian comparison theorem on Riemannian manifold. It refers to the gradient of distant function. Fix $x\in M$. Let $\rho(y)=d(y,x)$, and $r:I\to M$ is a minimal geodesic curve with ...
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1answer
33 views

'Large' closed subgroup

I am working through a paper in the field of differential geometry (Yang-Mills theory) and the author writes: 'We assume the Riemannian manifold $(M,h)$ admits a large closed subgroup $K$ of the ...
18
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1answer
285 views

Review on Riemannian Geometry

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...
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0answers
29 views

Example of locally symmetric spaces

A locally symmetric manifold is a manifold with parallel curvature tensor $\nabla R=0$. Can you give an example except spheres, projective spaces and hyperbolic spaces?
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1answer
27 views

a tangent vector which does not fall on any geodesic

Given a point on a manifold, is it possible that there is a tangent vector at that point which does not correspond to any local velocity of a some geodesic? That is in that direction no geodesic ...
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0answers
32 views

Proving that $\phi$ is orthogonal to the harmonic forms given $\int\phi \;d\mathrm{vol}$.

I want to prove that given a connected closed (compact without boundary) oriented Riemmanian manifold $(M,g)$, the condition $$\int_M \varphi \;\mathrm{d}\mathrm{vol}_g=0$$ implies that $\int \varphi ...
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1answer
24 views

Connection and curve

Let $\nabla$ be a connection on a Riemannian manifold and let the differential of a curve be given by $$c'(t)=c_1'(t)\partial_1 + c_2'(t) \partial_2.$$ Now I was wondering how we define ...
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41 views

Singer & Thorpe Theorem on the curvature of 4-dimensional Einstein spaces

I recently asked a question here about the paper "The curvature of 4-dimensional Einstein spaces." I got stuck again with the last theorem (2.2), where I get completely lost. They start the proof by ...
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0answers
28 views

Degenerate subspace

A null vector is a nonzero vector that is orthogonal to itself. If W is a subspace of V,let $W^{\perp}$ = [$v{\in}$ W : $v{\perp}$W]. $W^{\perp}$ is a subspace of V called W perp. A subspace W of ...
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1answer
21 views

Locality of tensors part of definition?

I am wondering whether linearity with respect to scalar functions $f \in C^{\infty}(M, \mathbb{R})$ is part of the definition of a tensor? Let me explain it by referring to the Riemann curvature ...
3
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0answers
32 views

Finding Riemannian metric from this geodesic

In a $d$-dimensional Riemannian manifold, given a geodesic equation $\gamma^i(t)=a^i\phi(tb^i),i\in 1\ldots d$, where $\phi:\mathbb{R}\rightarrow\mathbb{R}$ is an increasing function, $a^i,b^i$ are ...
0
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1answer
53 views

Volume element and orientabality

A volume element on an $n$-dimensional semi-Riemannian manifold $M$ is a smooth $n$-form $w$ such that $w(e_1,\cdots, e_n) = \pm1$ for every frame on $M$. How do I prove A semi-Riemannian ...
3
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1answer
82 views

Parallel Transport on a Cone

Suppose we have a cone and we wish to parallel transport a vector $w=(0,1,0)$ from along the curve $\alpha(s)=(\sqrt{2}/2 \cos(v\sqrt{2}),\sqrt{2}/2 \sin(v\sqrt{2}),\sqrt{2}/2)$ from $p=\alpha(0)$ to ...