I am asked first to to prove that an isometry ($\Phi$) preserves the Levi-Civita connection, this is: $$\Phi_*(\nabla_XY)=\nabla^*_{\Phi_*X}\Phi_*Y$$ where $\nabla^*$ refers to the Levi-Civita connetion of the codomain. Then I'm asked to do similarly for the Riemannian curvature. My problem is the next part: I am asked to show that if ($\Phi$) is now a local isometry, that the sectional curvature of a non-degenerate plane is preserved: $$K(\Pi)=K^*(d\Phi_p(\Pi))$$ And similarly for the Ricci tensor and the scalar curvature. My question is, why a local isometry only for the latter? I understand that a local isometry is more general than an isometry, so the way that this is set up seems to imply that the connection and the curvature tensor are not preserved by a local isometry. Is this true?
Maybe the question doesn't even make sense since the push forward of a vector field under a non-diffeomorphic mapping isn't well defined? But can't we calculate both the connection and the curvature tensor locally?