Complex sentence: Riemannian geometry studies Riemannian manifolds, smooth manifolds with a "Riemannian metric", a notion of a distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point.
Split into simpler sentences: Riemannian geometry studies Riemannian manifolds, smooth manifolds with a "Riemannian metric". This is a notional distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point.