Suppose that we have two smooth manifolds $M$ and $N$ and a smooth mapping $\phi : M \rightarrow N$. The Differential of that smooth mapping induces a bundle map $D\phi : TM \rightarrow TM$ between the tangential bundles.

We then have the pullback bundle $\phi^\star TN$ over the manifold $M$.

Let's now introduce Riemannian structures. Suppose that $M$ and $N$ are Riemannian manifolds with metrics $g^M$ and $g^N$ and the corresponding Levi-Civita connections $\nabla^M$ and $\nabla^N$.

1. Does the pullback connection $\phi^\star \nabla^N$ act as a connection on the pullback bundle $TN$?

2. Does the original connection $\nabla^M$ over $M$ act on the pullback bundle in any discernable way?

I believe the first question has a positive answer but I am in the dark on the second one. My motivation is whether one can take covariant derivatives of mixed tensor bundles such as $TM \otimes \phi^\star TN$ over $M$.

Suppose that we have two smooth manifolds $M$ and $N$ and a smooth mapping $\phi : M \rightarrow N$. The Differential of that smooth mapping induces a bundle map $D\phi : TM \rightarrow TM$ between the tangential bundles.

We then have the pullback bundle $\phi^\star TN$ over the manifold $M$.

Let's now introduce Riemannian structures. Suppose that $M$ and $N$ are Riemannian manifolds with metrics $g^M$ and $g^N$ and the corresponding Levi-Civita connections $\nabla^M$ and $\nabla^N$.

1. Does the pullback connection $\phi^\star \nabla^N$ act as a connection on the pullback bundle $TN$?

2. Does the original connection $\nabla^M$ over $M$ act on the pullback bundle in any discernable way?

3. What is a natural connection on "mixed" $TM \otimes \phi^\star TN$ over $M$?

I believe the first question has a positive answer but I am in the dark on the second and third one. My motivation is the third one in particular.