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Results tagged with riemannian-geometry
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user 13972
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
14
votes
Accepted
Examples of non isometric surfaces having the same curvature function
Yes, you can easily do this with surfaces of revolution: If you take a metric of the form $ds^2 = dr^2 + f(r)^2\,d\theta^2$, where $f$ is an odd function of $r$ satisfying $f'(0)=1$, then the Gauss c …
- 108k
7
votes
Accepted
Berger's theorem on Riemannian holonomy applied to the orthogonal frame bundle.
The answer is 'no' in general, when the dimension of the manifold is $n>2$. The holonomy group $H_x\subset\mathrm{O}(T_xM)$ could act transitively on the unit sphere in $T_xM$ and its identity compon …
- 108k
6
votes
Eigenforms for Laplacian on a non-flat two-torus
This is not so much an answer as a few remarks and a caution. If I understand your request correctly, I think that it is unlikely that you are going to find a truly explicit example.
First, let me r …
- 108k
3
votes
Accepted
2D-metric to diagonal form with determinant 1
Locally, this is always possible. Constructing such a coordinate system is equivalent to solving a first-order hyperbolic PDE system for two unknowns of two variables, so it always has local smooth s …
- 108k
6
votes
Accepted
Compact surface with genus$\geq 2$ with Killing field
Of course, there's always the Killing field $X\equiv0$. :)
Seriously, here's a different proof and an argument that addresses the 'suppose one leaves out a finite number of points' question:
Taking …
- 108k
11
votes
Accepted
Existence of normal and harmonic coordinates around a point
Yes. This follows from a standard fact about the Laplacian $\Delta_g$, (because it is a second-order elliptic operator): The fact is this:
If $u$ is a smooth function on an open neighborhood $U$ of …
- 108k
7
votes
Accepted
Orbits of Metrics under the Action of the Diffeomorphism Group
Answer: Not unless all of the $\lambda_i >0$ are equal to $1$ (or else, when $n=1$ and $\lambda_1$ and $\lambda_2$ are chosen so that the length of the curve $E_\lambda$ is $2\pi$; the condition for t …
- 108k
2
votes
Accepted
Existence of left-invariant metric on the cotangentbundle of homogeneous spaces?
No. The simplest example is $G = \mathrm{SL}(2,\mathbb{R})$ acting on $\mathbb{RP}^1 = G/P$, where $P$ is the (noncompact) subgroup of upper triangular matrices. It's easy to show that $G$ cannot no …
- 108k
4
votes
Accepted
geodesics on $G/K$ which are not the orbits of a 1-parameter subgroup of $G$
Perhaps I'm misunderstanding your question, but what about the following example?
Let $G = \mathrm{SO}(3)$ and let $K=\{e\}$ be the identity subgroup. Then $G/K = \mathrm{SO}(3)$ and $G_v = K$ for a …
- 108k
17
votes
Accepted
Is there some Riemannian manifold's version of Whitney theorem?
Yes, have a look at Robert Greene's book Isometric Embeddings of Riemannian and Pseudo Riemannian Manifolds, Volume 97 of Memoirs of the American Mathematical Society
Memoirs, 1970.
- 108k
5
votes
Accepted
Upper bound of derivative of exponential map
To get an upper bound of the kind you seek in general, you need a lower bound on $K$. Thus, for example, if you know that $K\ge -c^2$ on your surface, then you get
$$
\|\mathrm{d}(\exp_p)_v\|_{op} \ …
- 108k
13
votes
Accepted
Is the space of Levi-Civita connections convex
Here's a more specific approach that explains why you shouldn't expect this: For simplicity, I'll work in the 2-dimensional case, where it's probably the clearest. Let
$$
\omega = \begin{pmatrix}\om …
- 108k
7
votes
Accepted
Volume of $SO(n)\subset\mathbb R^{n^2}$, again
Maybe this will help: Regard $\mathrm{SO}(n)\subset M_{n,n}(\mathbb{R})$ as the set of $n$-by-$n$ matrices $a$ that satisfy ${}^ta\,a=\mathrm{I}_n$ and $\det(a)=1$. Then $\mathrm{SO}(n)$ is a smooth …
- 108k
15
votes
Accepted
Parallel transport as algebra isomorphism
It is a classic theorem in linear algebra that any ($\mathbb{R}$-linear) automorphism $\phi$ of the the ring $M_n(\mathbb{R})$ is inner, i.e., of the form $\phi(x) = axa^{-1}$ for some invertible $a\i …
- 108k
2
votes
Isometric embedding as a graph
As Anton points out, in the case that $q = \bar g - g$ has constant rank $k>0$, it is necessary that $K = \ker(q)\subset TM$ be integrable in order that $q = f^*h$ for some smooth map $f:M\to N$, wher …
- 108k