I am learning differential geometry and have a few questions on curvature. -- Background:
Gauss invented "Gauss curvature" to measure how surface curves.
Riemann gives an ingenious generalization of Gauss curvature from surface to higher dimensional manifolds using the "Riemannian curvature tensor" (sectional curvature is exactly the Gauss curvature of the image of the "sectional" tangent 2-dimensional subspace under the exponential map).
In modern textbooks on differential geometry, people usually first define the notion of connection, and then the Riemannian curvature tensor is expressed in terms of connection.
My questions are:
Are there some other notions of "curvature", besides Gauss curvature and Riemannian curvature as its generalization, which people have invented to measure how space curves?
In history, who first introduced the notion of connection to describe the Riemannian curvature tensor, and why is this idea natural?
Thank you very much!
