Riemannian Geometry

The study of Manifolds having a complete Riemannian Metric. Riemannian geometry is a general space based on the Line Element

$$ds=F(x^1, \ldots, x^n; dx^1, \ldots, dx^n),$$

with $F(x,y)>0$ for $y\not=0$ a function on the Tangent Bundle $TM$. In addition, $F$ is homogeneous of degree 1 in $y$ and of the form

$$F^2=g_{ij}(x)\,dx^i\,dx^j$$

(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.