Non-standard induced connection

[lavinia]

A hypersurface of Euclidean space obtains its Levi-Civita connection by orthogonal projection
of ordinary derivatives of vector fields in Euclidean space onto the hypersurface's tangent space.

Suppose rather than the unit normal, there is a non-zero transverse vector field and orthogonal projection is replaced by projection with respect to this vector field. What sort of geometry comes from this?

 

[Eynstone]

I'm afraid we need to calculate to say what happens ( or even whether a compatible geometry is induced by such a field). Could you share a worked-out example?

 

[lavinia]

I'm afraid we need to calculate to say what happens ( or even whether a compatible geometry is induced by such a field). Could you share a worked-out example?

It was just an idea but sure I will try to construct an example on the sphere and get back to you.

I wonder if these geometries are even torsion free.

 

[zhentil]

Hmm, how is this any different than pulling back the Levi-Civita connection for a different metric on R^3? I.e. if I declare the vector field to be orthogonal to the tangent space of the sphere, you would be getting a (perhaps different) Riemannian connection. Whether it's the Riemannian connection for the round metric on S^2 depends on whether the pullback metric is the same.

 

[zhentil]

In any event, I'm quite certain it would be torsion-free.