Riemannian connection basic question about computing

[samithie]

Does anyone have a good book or reference on computing riemannian connections. I'm looking at Do Carmo and can't find any examples. For ex. If X = y(d/dx) + x(d/dy) + w(d/dz) -z(d/dw), Y = z(d/dx) - w(d/dy) - x(d/dz) + y(d/dw), Z = w(d/dx) + z(d/dy) - y(d/dz) - x(d/dw) being the classical frame field on s3. How would I go about computing nabla_X(X), nabla_X(Y), nabla_X(Z) in terms of X,Y,Z where nabla is the levi civita connection. I would like to see one example if possible, or something similar. Thanks.

 

[Ben Niehoff]

The Levi-Civita connection satisfies two conditions. It is metric-compatible

$$\nabla_Z \big( g(X,Y) \big) = g(\nabla_Z X, Y) + g(X, \nabla_Z Y), \qquad \forall \, X,Y,Z$$
and torsion-free

$$\nabla_X Y - \nabla_Y X = [X,Y]. \qquad \forall \, X,Y$$
These two conditions should give you a system of linear equations which you can solve to obtain $\nabla_X X, \nabla_X Y,$ etc. In fact, it is possible to solve the system in general, without plugging in your definitions of the frame vectors. But it might be easier for you to do one example with the vectors you have.

 

[lavinia]

Does anyone have a good book or reference on computing riemannian connections. I'm looking at Do Carmo and can't find any examples. For ex. If X = y(d/dx) + x(d/dy) + w(d/dz) -z(d/dw), Y = z(d/dx) - w(d/dy) - x(d/dz) + y(d/dw), Z = w(d/dx) + z(d/dy) - y(d/dz) - x(d/dw) being the classical frame field on s3. How would I go about computing nabla_X(X), nabla_X(Y), nabla_X(Z) in terms of X,Y,Z where nabla is the levi civita connection. I would like to see one example if possible, or something similar. Thanks.

The connection will have an simple form in geodesic polar coordinates. For the sphere this is an easy computation. The Poincare disc should also be easy.