# Geometric way to view affine connection and parallel transport.

Given a parametrized curve $$\gamma$$ on a manifold $$M$$ with metric $$g$$ and some affine connection on it, we can transport vectors of tangent spaces $$T_{p} M$$ and $$T_{q}M$$ to each other (when $$p, q \in \gamma$$). Then scalar product of vectors (w.r.t. $$g$$) is preserved under this transport and we can see it algebraically just because of Leibniz rule.

The question is: Can we see the fact that scalar product is preserved in a more geometric way?

Any notes on geodesics will also be appreciated.

One may consider $$M$$ as a submanifold in some $${\mathbb R}^n$$ and assume $$X, Y$$ are two vector fields on $$M$$. Then the Levi-Civita connection (at $$x\in M$$) $$\nabla_XY$$ is just taking the ordinary Euclidean directional derivative for $$Y$$ in the $$X$$ direction, then project to the tangent space $$T_x M$$. Euclidean derivative preserves the scalar product (by calculus), and this is still true after projecting to $$T_x M$$.