User kangdon - MathOverflow

Higher derivatives than Jacobi fields.

2011-09-15T05:37:32Z

Hi,

the first and second derivatives of the distance function (either the full $d:M\times M\to \mathbb{R}$ function or the $d(p,\cdot):M\to \mathbb{R}$ function) as well as the derivative of the exponential map (again both of the full $\exp:TM\to M$ map and of the map $\exp_p:T_pM\to M$) may be computed with the aid of Jacobi fields, i.e, solutions to Jacobi's equation.

I have a scenario where I need second derivatives of the `full' exponential map $\exp:TM\to M$. That is, denoting the pushforward of a differentiable map by a '$ _*$', I would like to know a thing or two about $\nabla_X\exp_\ast\mathcal{V}$ (where $\mathcal{V}\in TTM$ and $X$ is an appropriate vector field). In particular, I think I will require some comparison techniques analagous to those for Jacobi fields (e.g Rauch's comparison theorems).

Can anyone point me in the right direction?

Cheers,

Mat

Answer by kangdon for What is torsion in differential geometry intuitively?

2012-08-28T01:06:52Z

One more interpretation: The torsion is the curvature of the smooth functions (as a vector bundle over your manifold).

Answer by kangdon for Which math paper maximizes the ratio (importance)/(length)?

2011-11-08T03:32:04Z

Perelman's ``Proof of the soul conjecture of Cheeger and Gromoll.'' J. Differential Geom. 40 (1994), no. 1, 209–212,

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214455292

is, at 3 pages (plus a paragraph of remarks), a favourite of mine, although it has some pretty tough competition here.

Applications of geometric evolution equations.

2011-07-08T05:10:13Z

Hi everybody,

I'm looking for applications of geometric evolution equations such as the Ricci flow and the extrinsic flows by Gauss and mean curvature. Applications other than topological applications such as geometrization are what I'm looking for and, even better if the application is to a non-mathematical area, such as Mullins' derivation of mean curvature flow as a model for the motion of `grain boundaries'.

Thanks,

Answer by kangdon for Roadmap to learning about Ricci Flow?

2011-07-03T15:08:49Z

In would recommend the book `The Ricci Flow in Riemannian Geometry' by Ben Andrews and Chris Hopper, which is available for download here:

http://maths.anu.edu.au/~andrews/publications.html

The book is suited to an honours/graduate student with a good background in Riemannian geometry. It develops Hamilton's Ricci flow from the ground up leading to Brendle and Schoen's proof of the differentiable sphere theorem and also provides a very good overview of the required geometry in the first chapter.

Suggestions for teaching advanced high school students

2011-06-26T04:56:45Z

Hi all,

I'm a grad student and just joined a mentoring program in which I will visit a group of advanced year ten high school students (around 16 years old) from a group of schools in the area. I don't quite know where they're at just yet. I assume they've done Euclidean geometry, analytic algebra are just getting used to differential calculus. I'm fairly free to do what I want with them short of cruel and unusual punishment (such as making them read Gilbarg and Trudinger). They meet weekly and I had in mind meeting them every two weeks and going through a shotgun summary of what maths is all about, with an eye for a historical development. Of course, I want it to be fun! I have a semseter, so that makes about 6 or 7 two hour meetings.

To make the question precise: What do the potential answerers believe should not be left out of such an introduction?

Thanks,

Mat

Hessian of the distance function and Cartesian products of manifolds

2011-04-20T01:02:39Z

Hi,

I have a function $Z:M\times M\to \mathbb{R}$ defined by two points on the manifold $M$. $M$ is a submanifold of a Riemannian manifold $N$ and $Z$ involves the Riemannian distance function $d$ on $N$. I would like to compute the first and second derivatives of $Z$ (perhaps with respect to some coordinates $(x^i,y^i)$ on $M\times M$.) One can compute the derivatives of $d$ at $(x,y)$ by considering a geodesic variation (in $N$) of the geodesic $\gamma$ joining $x$ and $y$. One gets:

$$(D|_{(x,y)}d)(u,v)=\big< v,\gamma'(d(x,y))\big> -\big< u,\gamma'(0)\big>$$

where $(u,v)\in T_xM$

I haven't yet computed the Hessian but the same technique should work. My problem is that $Z$ also involves the vector field $w:M\times M\to N$ defined by $w(x,y)=\gamma'(0)$, where $\gamma$ is as above. In fact, the term I have is $\big< w,\nu_x\big>$, where $\big<\cdot, \cdot\big>$ is the metric on $N$ and $\nu_x(x,y)$ is the outer unit normal at $x$. Note, though that $$\big< w,\nu_x\big> = -(Dd)(\nu_x)=-(Dd)(\nu,0)$$

Any ideas on how I can compute the derivatives of this expression? The problem seems to be that my vector fields are sections of the bundle $TM$ over the product $M\times M$. So perhaps I should be asking what the the connection should look like on $\Gamma(M\times M,TM)$?

Note also that in $N=\mathbb{R}^n$ the answer is easy since $w=(y-x)/|y-x|$

Thanks.

Differential Geometry/General Relativity Computer Algebra

2011-01-27T03:18:10Z

Hi,

could anybody recommend a CAS suited to DG/GR applications such as computation of connection coefficients or generating (and possibly solving) PDEs for, for example, an unknown metric of given curvature. Oh, and compatible Linux (I'm using Maple through wine but am having myriad problems. Also tried Maxima but I don't think it has a PDE solving tool.)

Thanks,

Mat

Retarded coordinates on (flat) spacetime

2011-01-20T11:36:22Z

Hi,

I'm trying to construct some coordinates on Minkowski spacetime based on a world line, $C$, ($\dot{C}\cdot\dot{C}=-1$) and forward light cone. I want the "time" coordinate of a point, $p$, to be the "retarded time", i.e the time $t(p)$ at the (unique when it exists) point $C(t(p))$ on $C$ joining $p$ by a null geodesic $\gamma$ (on the forward light cone of $C(t(p))$: $\gamma'\cdot\dot{C}<0$). I want the "distance" coordinate to then be the `light distance': $r(p)=-(p-C(t(p)))\cdot\dot{C}(t(p))$ and some angular coordinates defined by the "direction" (possibly via a projection onto an instantaneous 3-space) of $\gamma$ at $C$.

The problem I'm having is constructing the metric. Say I start with the vector field $\dot{C}$ and a triad ("forward") of linearly independent null vectors $N_i$ (so that $g(N_i,N_j)\neq 0$). If I Fermi-Walker transport this triad along $C$: \begin{equation} \nabla_{\dot{C}}N_i=(N_i\cdot \ddot{C})\dot{C}-(N_i\cdot \dot{C})\ddot{C}\quad (\ddot{C}=\nabla_{\dot{C}}\dot{C}) \end{equation} then they remain a null triad and do not rotate (I think). I could then choose to parallel translate $\dot{C}$ and $N_j$ along the cone to obtain a local basis field.

But seeing as the $N_i$ are null I'm not quite sure how to construct the metric (that is, assuming my previous steps are ethical).

The reason I would like these coordinates is to study solutions to Maxwell's equations \begin{equation} \mbox{d}F=0\qquad \mbox{d}\ast F = j \end{equation} and hopefully obtain something like the Liénard-Wiechert potential in the language of exterior calculus. So any references would also be nice (I've tried the standard texts, i.e. Jackson and Rohrlich).

Cheers,

Mat L

Action of the group of isometries on a manifold

2010-07-08T09:17:46Z

Hi guys,

I am able to prove that any symmetric manifold is complete (Consider a local geodesic and use the symmetry to flip it, effectively doubling the length of the geodesic, ad infinitum). I want to use a similar procedure to prove that a manifold whose isometries act transitively is complete, i.e there is always an isometry which maps the start point of a local geodesic to its end point, preserving the geodesic. I am, however, unable to ensure that it is not `rotated' in the process, i.e I want the pushforward of the initial tangent, by the isometry, to be the final tangent, ensuring the resultant doubled geodesic is smooth.

My Lie group theory is a bit scratchy but I assume there is a method which allows me to construct the correct pushforward using only transitivity.

Any ideas would be great,

regards,

Topological results from geometry

2010-03-25T01:41:13Z

Hi people,

I'm interested in results, such as the Gauß-Bonnet theorem, Fàry-Milnor theorem or classification theorems for manifolds, which give topological properties from geometric considerations. Can anyone recommend some good texts? In particular I'd like to see a nice proof of Fàry-Milnor and of the theorem of turning tangents (total curvature of an imbedded plane curve is $2\pi$).

thanks

Comment by kangdon

2012-03-05T00:37:57Z

It's fun to compute an explicit example, (i.e, the sphere with it's standard metric). This makes it easier to see what Deane means.

Comment by kangdon

2011-12-06T04:13:20Z

Hi, this probably isn't much help. Tensors (of rank two) satisfying $\nabla_Xh(Y)-\nabla_Yh(X)=0$ are also called Codazzi tensors. There are many of them (e.g $g$ or the second fundamental form) and many people find them interesting. For example: According to arxiv.org/abs/1111.7002 Berger-Ebin (projecteuclid.org/…) proved that a constant trace Codazzi tensor on a compact manifold with non-negative sectional curvature must be parallel ($\nabla h=0$).

Comment by kangdon

2011-10-24T09:57:09Z

'$\backslash$dpts' should read '...'

Comment by kangdon

2011-10-24T09:54:32Z

This is an inappropriate forum for this question, particularly if it's an assignment question! In any case, your problem is the formula $(\nabla_U(\nabla X))(V)=\nabla_U(\nabla_V X)$, which is not true. When differentiating a tensor, you `commute with contractions': $$ (\nabla_X T)(U,V,\dots)=\nabla_X(T(U,V,\dots))-T(\nabla_XU,V,\dots)-T(U,\nabla_XV,\dpts)+\dots $$

Comment by kangdon

2011-10-01T01:59:02Z

Is there a similar statement of sharpness?

Comment by kangdon

2011-09-17T09:30:59Z

A small correction: $\mathcal{V}$ should rather be a section of $TTM$.

Comment by kangdon

2011-09-16T12:37:56Z

Theoretically, so long as you have access to a library and journals it is not impossible to undertake your own research, as Peter O'Sullivan has shown: maths.anu.edu.au/events/Osullivan60. Unfortunately, I imagine this would be hugely more difficult than having guidance from mentors and a community. Luckily mathematicians are generally pretty friendly, as per the helpful comments already given.

Comment by kangdon

2011-09-15T13:11:15Z

Thanks Deane, I figured as much but after a couple of tentative attempts at computing it I realised I was in for some work and hoped there'd be some literature someone could point to. As you say, surely it's been done before to a similar end. I suppose it will be a good exercise.

Comment by kangdon

2011-07-08T12:56:38Z

Most interesting. Thanks Joseph!

Comment by kangdon

2011-07-08T01:13:10Z

@Adam: Only if $n=7$

Comment by kangdon

2011-06-26T07:55:07Z

Good point KConrad. I based my assumptions on my own experience. And guessed incorrectly. The syllabus doen't include differentiation until year 11. It is an Australian school. The idea is to get a feel for where they're at during the first meeting. But I wanted to start considering some things to talk about earlier. As mentioned, they're a group of students from different schools who are top of their respective classes and quite interested in mathematics. My goal is to ensure they stay interested in maths and to teach them what maths is `really' about (Something the syllabus doesn't do)

Comment by kangdon

2011-04-18T00:54:38Z

(Possibly of positive curvature)

Comment by kangdon

2011-04-18T00:53:42Z

The Hadamard-Cartan comparison theorem says that any complete Riemannian manifold of non-positive sectional curvature has no conjugate points. Does anyone know of an `easily-pictured' example of a compact (boundaryless) manifold without conjugate points?

Comment by kangdon

2011-04-18T00:53:26Z

@unknowngoogle: I believe The `counterexample' to the proof is in $\mathbb{R}^4$.

Comment by kangdon

2011-01-28T00:29:58Z

I think I'm in the same boat as you José. I use Maple and MatLab just unoften enough that I need to relearn everything every time. Thanks for the links.