Can I have two connections on a Riemannian Manifold compatible with two different metrics and are torsion free(Levi Civita)?. If yes, then are the related or independent?
I am particularly concerned with the pullback connection.
If $\phi:(M,g,\nabla)\rightarrow (N,\tilde{g},\tilde{\nabla})$ is a diffeomorphism and $(\phi^*\tilde{\nabla})_XY=\phi_*^{-1}(\tilde{\nabla}_{\phi_* X}(\phi_* Y))$ is the pullback connection on the pullback bundle over M compatible with the pullback metric $\phi^*\tilde{g}$.
Are these two connection same or $(\phi^*\tilde{\nabla})_XY-\nabla_XY=D(X,Y)$? Is the Pullback bundle same as the bunfle on M? What happens if I take M=N but keep the Riemannian structure different?
Please forgive me If my question is not well framed as I am currently studying this.