Flatness of a manifold (or a connection) Suppose we have an $n$-dimensional manifold $S$ (with a global coordinate system) with a metric $g$ and a connection $\nabla$ with connection coefficients (Christoffel symbols) $\Gamma_{i,j}^k$ given. Suppose that the $\nabla$-geodesic connecting any two points of the manifold completely lies in $S$. Can we then say that $S$ must be flat with respect to the given connection? I am not able to straightaway show that $(\Gamma_{i,j}^k)_p = 0$ at all points $p$ of $S$.

I think you are misunderstanding what a flat (affine) connection is: It is a connection on a manifold $M$ such that at each point of $M$ there exists a coordinate system with zero Christoffel symbols (vanishing depends heavily on which coordinates you use). Equivalently, a connection is flat if it has zero curvature. Equivalently, it is flat if parallel transports along contractible loops are identity maps, etc. This will be explained in any Riemannian geometry textbook; my favorite is do Carmo's "Riemannian Geometry" (chapters 0 through 4). Or use Petersen's "Riemannian Geometry".