The study of Manifolds having a complete Riemannian Metric.
Riemannian geometry is a general space based on the Line Element
References
Besson, G.; Lohkamp, J.; Pansu, P.; and Petersen, P. Riemannian Geometry.
Providence, RI: Amer. Math. Soc., 1996.
Buser, P. Geometry and Spectra of Compact Riemann Surfaces. Boston, MA: Birkhäuser, 1992.
Chavel, I. Eigenvalues in Riemannian Geometry. New York: Academic Press, 1984.
Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.
Chern, S.-S. ``Finsler Geometry is Just Riemannian Geometry without the Quadratic Restriction.''
Not. Amer. Math. Soc. 43, 959-963, 1996.
do Carmo, M. P. Riemannian Geometry. Boston, MA: Birkhäuser, 1992.