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.

© 1996-9

1999-05-25