Question: Pulling a connection back

September 17 2011 by
false
Maple 14

2

Hi!

I'm studying submanifolds of (semi)-riemannian and trying to look at some examples with Maple using it's great DifferentialGeometry package. But there's one thing I can't figure out. How can I define a pullbacked tangentbundle along the embedding map? Let me be more specific:

I have a manifold S, a manifold M and a map i: S -> M.

I realize this in Maple with: DGsetup( [...], M ); DGsetup( [...], S); i:=Transformation(S,M,[...]);

There is a metric defined on M, let's call it

G:=evalDG(... dx &s dx ...);

The induced metric on S is g = i^* G, which I can calculate by calling:

g:=Pullback(i,G); or equivalent: g:=PushPullTensor(i,G);

There are connections (the Levi-Civita connections) defined on M and S, which I can construct with

CG:=Connection(Christoffel(G)); Cg:=Connection(Christoffel(g));

In the theory of submanifolds, there is a very important relation between CG and Cg. It is called the second fundamental form. It is given by covariantly differentiating the pushforward basis fields:

X:=Pushforward(i,DGinfo("FrameBaseVectors");

But here is the problem, I can't solve with Maple. The pushforward fields are vectorfields along the embedding map i, i.e. they are linear combinations of the FrameBaseVectors of M with coefficients defined on S! Covariant differentiation of those is given by applying the chain rule, i.e. differentiating the coefficients with the exterior derivative and applying the CG to the FrameBaseVectors. But I can't figure out, how to do that with maple.

Maybe my "Ansatz" is wrong, and I had to define S as a Vectorbundle

DGsetup([coordinates of S],[coordinates of M],E)?

I can define the transformation i now, by mapping the base coordinates to the frame coordinates through "the old rule of i", but what do I set the new base coordinates to? To 0? Also I can't calculate the connection through the Christoffel method anymore :(

Maybe somebody of you has a good idea how to proceed?