3
votes
1answer
59 views

Is a Riemannian metric a $2$-form?

In Lee's Riemannian Manifolds; An introduction to Curvature, he defines a Riemannian metric as an element of $\Gamma(T^2_0M)$, a $(2,0)$-tensor. Is this the same thing as a $2$-form? Is there a ...
2
votes
0answers
59 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
7
votes
1answer
137 views

Uniqueness of curve of minimal length in a closed $X\subset \mathbb R^2$

Suppose $X$ is a simply connected closed subset of $\mathbb R^2$. Let $a,b$ belong to $X$. Is it true that there is at most one curve inside $X$ from $a$ to $b$ such that the length of the curve is ...
0
votes
0answers
55 views

Isometries of a general metric

For a general (pseudo-)Riemannian manifold, i.e. in which the interval $ds$ can be written $ds^2 = g_{ab}\,dx^a \,dx^b$, is there a general prescription for finding the group of isometries- by ...
6
votes
0answers
98 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha ...
1
vote
1answer
56 views

Superspace as the Hilbert Space for Quantum Gravity

This is a question I've asked in physics.stackexchange, but have obtained no answers: Let $\mathcal{A}$ be the Ashtekar connection. Since $^{(3)}g_{AB}=i\frac{\delta}{\delta\mathcal{A}^{AB}}$ (see R. ...
0
votes
0answers
31 views

Reality condition on a metric

I'm studying a co├Ârdinate-transformation on a 2n-dimensional real manifold, where I can locally define the co├Ârdinates as $(x^1,...,x^{2n})\in\mathbb{R}^{2n}$, and transform them to: ...
0
votes
0answers
63 views

Hadamard space: property of the Busemann function

I have a question about a property of Busemann functions on Hadamard spaces. Let $X$ be a complete CAT($0$) space. If $r:[0, \infty) \to X$ is a geodesic ray, and $x\in X$ the Busemann function is ...
3
votes
1answer
76 views

D'Alembertian $\Box$

This question has to do with the D'Alembertian operator on a general manifold with a metric $g_{\mu\nu}$. I understand that the definition of the D'Alembertian is $$\Box \phi\equiv ...
3
votes
2answers
90 views

On the definition of a geodesic in a metric space

I am interested in the definition of geodesics in metric spaces. A definition which seems reasonable to me is that a geodesic should locally be a distance minimizer. Wikipedia ...
1
vote
1answer
58 views

Limit of Riemannian metrics on the disk.

I'm working through Burago, Burago and Ivanov's book A Course in Metric Geometry and I'm trying to solve the following excercise: If we denote by $D^2$ the standard unit ball in $\mathbb{R}^2$, then ...
2
votes
0answers
44 views

Finding an isometry given the distance between points

If $A,B,C,D \in \mathbb{R}^n$, such that $d(A,B)=d(C,D)$, then exists an isometry $f:\mathbb{R}^n\rightarrow \mathbb{R}^n $, such that $f(A)=C, f(B)=D$. Thanks a lot for the help.
5
votes
1answer
114 views

Metric on Riemannian manifolds

Why is it necessary to consider taking the infimum over the lengths of all piece-wise smooth curves while defining the distance function on a Riemannian Manifold instead of just taking the infimum ...
4
votes
2answers
138 views

Distance metric for (infinite) lines in 3D

I would like a metric $d(\;)$ between pairs of (infinite) lines in $\mathbb{R}^3$, with these properties: If two lines $L_1$ and $L_2$ are parallel and separated by distance $x$, then $d(L_1,L_2) = ...
1
vote
1answer
48 views

One Point Derivations on locally Lipschitz functions

Let $A$ be the algebra of $\mathbb{R}\to\mathbb{R}$ locally Lipschitz functions. What is the vector space of derivations at $0$? The proof that for continuous functions there aren't really any doesn't ...
3
votes
1answer
82 views

geodesic metric

I'm trying to prove that the line segment is the minimizer of the distance $$d(x,y)=\inf l(\gamma),$$ where $x,y\in X$, $X$ is a Banach space, $\gamma$ is a path from $x$ to $y$ and ...
1
vote
0answers
23 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
1
vote
1answer
41 views

Existence of Lipschitz reparametrization

Suppose we are given a continuous path, $$\gamma:[0,1]\rightarrow (X,d)\text{,}$$ in a metric space $(X,d)$. When we deal with differentiable enough paths in Riemann manifolds we can give a ...
1
vote
1answer
34 views

Metric spaces and curvature

Suppose that it is given that the Riemann curvature tensor in a metric space of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the ...
1
vote
1answer
98 views

Coordinate transform

Can anyone see what transformation $$r\to f(r)$$ transforms $$\exp(2\phi(r))(dr^2+r^2d\theta^2)$$ to $$df^2+\sinh^2(f)d\theta^2$$? I there a systematic way to attack such a problem -- rather than just ...
1
vote
0answers
22 views

A better way to see this relation concerning Ricci tensor components

If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
2
votes
1answer
352 views

Prove that two normed linear spaces are equivalent as metric spaces if and only if the norms are equivalent?

We have the two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on the vectorspace V. They're equivalent if there exists a $k>0$ and $K>0$ so that $k\|\cdot\|_a\le\|\cdot\|_b\le$ K$\|\cdot\|_a$ for all ...
8
votes
1answer
118 views

Hyperbolic diameter of Amsler's surface

I've recently learned about Amsler's surface, a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of ...
11
votes
1answer
333 views

Metric in riemannian Manifold

Let $(M,g)$ be a riemannian Manifold, we can use the metric $g$ to obtain a metric $d_g:M\times M\to \mathbb{R}$. I ask for a kind of converse, we can start with a metric $d:M\times M\to \mathbb{R}$ ...
0
votes
1answer
169 views

variation problem of constrained area and minimized distance

$$c=\int_{x_1}^{x_2}f_{gr}(x)\;dx$$ The integral is a time-like curve between $x_1$ and $x_2$ and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and ...
1
vote
2answers
388 views

Index notation and differentiation

Let $x_i$ such that $i=1,2,\ldots,n$, and $\vec{x}=(x_1,\ldots,x_n)$ Define $$A:= M_{ij}(\vec{x})\dot{x}^i\dot{x}^j$$ where Einstein summation applies. Also, $M$ is symmetric and invertible -- a ...
3
votes
1answer
163 views

Coordinate change for metrics

I am rather confused by the idea of "geodesic polar coordinates", so I hope someone would kindly explain it to me. As far as my understanding goes, given a Riemannian metric ...
7
votes
1answer
339 views

Metric on $\Bbb{R}^n$ which comes from a continuous function

I've been struggling to prove the following fact for some time now, and I didn't manage to do so. Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros ...
3
votes
2answers
635 views

Tori and metrics

I have been doing some reading on tori. What I can make out of it is that a torus can be equipped with different metrics -- locally Euclidean or as an embedded surface. It is said however that the ...
1
vote
0answers
95 views

Distinct metrics on a manifold

I'm trying to understand basic differential geometry (my background is in mathematical logic), and I'm having a bit of difficulty with a basic point. Frequently we want to consider the set of metrics ...
1
vote
1answer
90 views

difference between locally equivalent and uniformly equivalent for metrics on a manifold

Suppose $M$ is a manifold, and two metrics $g_0$ and $g_1$ are given on $M$. What is the difference between the assertion "$g_0$ and $g_1$ are locally equivalent" and "$g_0$ and $g_1$ are uniformly ...