User van abel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:54:30Z http://mathoverflow.net/feeds/user/18801 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89061/whats-the-relationship-between-the-riemannian-metric-and-jacobi-field What's the relationship between the riemannian metric and Jacobi field? van abel 2012-02-21T00:37:10Z 2012-02-21T13:59:11Z <p>I encounter to the question in reading the following Excise:</p> <blockquote> <p>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 $$f(r)=\begin{cases} \sin(\sqrt{Kr^2})/\sqrt{K},&amp;\text{if }K>0; r,&amp;\text{if }K=0; \sinh(\sqrt{-Kr^2})/\sqrt{-K},&amp;\text{if }K&lt;0. \end{cases}$$</p> </blockquote> <p>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 $W(t)\bot\dot\gamma(t)=T(t)$.</p> <p>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?</p> <p>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.</p> http://mathoverflow.net/questions/79464/perpendicular-in-conformal-disk-model Perpendicular in conformal disk model van abel 2011-10-29T13:41:19Z 2011-10-30T03:45:13Z <p>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?</p> <p><img src="http://www.freeimagehosting.net/newuploads/21264.jpg" alt="Figure"></p> <p>in the above figure, <strong>disk A</strong> is the conformal model disk. <strong>J</strong> is the given point, and <strong>CD</strong> is the given line; I asked how to get the line <strong>GF</strong>, such that <strong>J</strong> is on it and <strong>GF</strong> perpendicular to <strong>CD</strong>.</p> http://mathoverflow.net/questions/79167/is-there-a-similar-formula-in-spherical-and-hyperbolic-geometry-as-euclidean-geom Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? van abel 2011-10-26T15:14:14Z 2011-10-26T22:42:55Z <ol> <li><p>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?</p></li> <li><p>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.</p></li> </ol> http://mathoverflow.net/questions/103956/did-the-real-number-system-unique Comment by van abel van abel 2012-08-06T02:50:14Z 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 &lt;em&gt;ordered&lt;/em&gt; field $R$ is called real field, if ... http://mathoverflow.net/questions/89061/whats-the-relationship-between-the-riemannian-metric-and-jacobi-field Comment by van abel van abel 2012-02-22T11:54:08Z 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. http://mathoverflow.net/questions/89061/whats-the-relationship-between-the-riemannian-metric-and-jacobi-field Comment by van abel van abel 2012-02-21T00:41:48Z 2012-02-21T00:41:48Z This question is firstly post on StackExchange:<a href="http://math.stackexchange.com/questions/111504/whats-the-relationship-between-the-riemannian-metric-and-jacobi-field" rel="nofollow" title="whats the relationship between the riemannian metric and jacobi field">math.stackexchange.com/questions/111504/&hellip;</a>, after I read the site meta, I think is more prefer to post it here. http://mathoverflow.net/questions/79464/perpendicular-in-conformal-disk-model Comment by van abel van abel 2011-10-30T03:56:48Z 2011-10-30T03:56:48Z the question continued at [link stackexchange][2]. [2]: <a href="http://math.stackexchange.com/questions/77084/perpendicular-in-conformal-disk-model" rel="nofollow" title="perpendicular in conformal disk model">math.stackexchange.com/questions/77084/&hellip;</a> http://mathoverflow.net/questions/79464/perpendicular-in-conformal-disk-model Comment by van abel van abel 2011-10-29T13:43:53Z 2011-10-29T13:43:53Z May I ask why I can't use the <b>img tag</b> in my question? http://mathoverflow.net/questions/79167/is-there-a-similar-formula-in-spherical-and-hyperbolic-geometry-as-euclidean-geom Comment by van abel van abel 2011-10-27T03:02:38Z 2011-10-27T03:02:38Z Thanks, That is reasonable, I will do that in my following questions. http://mathoverflow.net/questions/79167/is-there-a-similar-formula-in-spherical-and-hyperbolic-geometry-as-euclidean-geom Comment by van abel van abel 2011-10-26T16:10:31Z 2011-10-26T16:10:31Z Is that illegal to a same question on two site? if it is, I owe an apologize.