I am a little confused by an idea suggested to me: putting a connection on a sphere doesn't specify a metric geometry - it remembers notions like straightness of paths, but not length of paths.
Let me ask a specific question.
Let $\nabla$ be a connection on the smooth sphere $S^2$ (a topological manifold with a smooth atlas). Provided the connection $\nabla$ spherically symmetric (I capriciously define this to mean that it pulls back to itself for all elements in some conjugacy class of $SO(n)$ in the full diffeomorphism group of the zero set $x^2 + y^2 + z^2 = 1$ in $R^3$), torsion-free, and has constant scalar curvature $1/2(1/r^2)$, can I conclude that the connection is the Levi-Civita connection of the round sphere of radius $r$ with the usual spherical metric?
Thanks - I am just trying to understand better exactly what information is contained in the connection / covariant derivative.