User van abel - MathOverflow

What's the relationship between the riemannian metric and Jacobi field?

2012-02-21T00:37:10Z

I encounter to the question in reading the following Excise:

Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ be the (geodesic) polar coordinate. Prove that, if $M$ is a space form, i.e., with the constant (sectional) curvature $K$, then $g$ has the following expression (by Gauss's Lemma): $$ \newcommand{\rd}{\mathrm d} \rd s^2=(\rd r)^2+(f(r))^2h_{ij}(\theta)\rd\theta^i\rd\theta^j, $$ where the $m-1$-dimensional metric $(\rd\sigma)^2=h_{ij}(\theta)\rd\theta^i\rd\theta^j$ has constant sectional curvature $1$, and $$ f(r)=\begin{cases} \sin(\sqrt{Kr^2})/\sqrt{K},&\text{if }K>0; r,&\text{if }K=0; \sinh(\sqrt{-Kr^2})/\sqrt{-K},&\text{if }K<0. \end{cases} $$

I already know that the Jacobo-field $J$ ($J(0)=0$, $|\dot J(0)|=1$ and $J$ is perpendicular to the base geodesic curve $\gamma(t)$) for manifold with constant sectional curvature $K$ is $$ J(t)=f(t)W(t), $$ where $f(t)$ is defined as above and $W(t)$ is a unit parallel vector filed along $\gamma$ with $W(t)\bot\dot\gamma(t)=T(t)$.

So my question is that: can we give a proof that based on the Jacobi-field? If not, what's the relation between the two problem?

In fact, I don't know how the solve the first problem, and try to use Jacobi-field to give a proof without no results.

Perpendicular in conformal disk model

2011-10-29T13:41:19Z

In Euclidean Geometry, we know that from a given point there is an unique line perpendicular to a given line. In conformal disk model, can we do the same thing? Or, more exactly, did there exists such a (geodesic) line, if exists, are they unique? if not, when?

in the above figure, disk A is the conformal model disk. J is the given point, and CD is the given line; I asked how to get the line GF, such that J is on it and GF perpendicular to CD.

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?

2011-10-26T15:14:14Z

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, i.e., $$ S_{\Delta}=\frac{1}{2}a.h, $$ where $a$ is the length of base and the $h$ is the height. Is there a similar formula in Spherical and hyperbolic spaces?
In Euclidean plane, we know that the cosine law says that (suppose $\alpha$, $\beta$, $\gamma$ are the angles, and $a, b, c$ are the lengths opposite $a, b, c$ respectively.) $$ \cos\gamma=\frac{a^2+b^2-c^2}{2ab}. $$ Then is there a analogue in spherical and hyperbolic geometry? I have noted that the First and Second Cosine law are not so clearly relevant with this formula.

Comment by van abel

2012-08-06T02:50:14Z

Thanks for Yuan's comments, maybe my definition of real fields are not correctly for my question, and maybe the definition should be: A <em>ordered</em> field $R$ is called real field, if ...

Comment by van abel

2012-02-22T11:54:08Z

Thanks for all the above suggestions. At first, I just suspected that there would be a solution by using the Jacobi fields (which was comformed by Deane Yang), believing that it should be more efficient, I just stop here, do not want to have a try by the definition of section curvature. This maybe a problem for me, especially, as a beginner. Now, I realize that, sometimes the clumsy caculation is necessary. Of course, I will have a try right now, and wish that I will work it out before exhausting my effort.

Comment by van abel

2012-02-21T00:41:48Z

This question is firstly post on StackExchange:math.stackexchange.com/questions/111504/…, after I read the site meta, I think is more prefer to post it here.

Comment by van abel

2011-10-30T03:56:48Z

the question continued at [link stackexchange][2]. [2]: math.stackexchange.com/questions/77084/…

Comment by van abel

2011-10-29T13:43:53Z

May I ask why I can't use the img tag in my question?

Comment by van abel

2011-10-27T03:02:38Z

Thanks, That is reasonable, I will do that in my following questions.

Comment by van abel

2011-10-26T16:10:31Z

Is that illegal to a same question on two site? if it is, I owe an apologize.