Riemannian Geometry

The study of manifolds having a complete Riemannian metric. Riemannian geometry is a general space based on the line element

with  for  a function on the tangent bundle . In addition,  is homogeneous of degree 1 in  and of the form

(Chern 1996). If this restriction is dropped, the resulting geometry is called Finsler geometry.

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.