In a first course aiming to introduce differentiable manifolds as the spaces on which do calculus, you could give to the students the notion of connection at least on vector bundles.

In order to reflect on the reason for this choice, I report the words of S.S.Chern closing the introduction of "Global Differential Geometry, Studies in Math.vol.27, MAA":

The Editor is convinced that the notion of a connection in a vector bundle will soon find its way into a class on advanced calculus, as it is a fundamental notion and its applications are wide-spread. His chapter, "Vector Bundles with a Connection," hopefully will show that it is basically an elementary concept.

In a first course aiming to introduce differentiable manifolds as the spaces on which do calculus, you could give to the students the notion of connection at least on vector bundles.

In order to reflect on the reason for this choice, I report the words of S.S.Chern closing the introduction of Global Differential Geometry, MAA Studies in Math.27, 1989:

The Editor is convinced that the notion of a connection in a vector bundle will soon find its way into a class on advanced calculus, as it is a fundamental notion and its applications are wide-spread. His chapter, "Vector Bundles with a Connection," hopefully will show that it is basically an elementary concept.