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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,...,x^n;dx^1,...,dx^n),

with F(x,y)>0 for y!=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^idx^j

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

See also

Non-Euclidean Geometry, Riemannian Metric

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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.

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Riemannian Geometry

Cite this as:

Weisstein, Eric W. "Riemannian Geometry." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannianGeometry.html

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