The study of manifolds having a complete Riemannian metric. Riemannian geometry is a general space based on the line element d s = F(x^1, ..., x^n ;d x^1, ..., d x^n), with F(x, y)>0 for y!=0 a function on the tangent bundle T M. In addition, F is homogeneous of degree 1 in y and of the form F^2 = g_(i j)(x) d x^i d x^j (Chern 1996). If this restriction is dropped, the resulting geometry is called Finsler geometry.

non-Euclidean geometry | Riemannian metric