Decomposition theorem from Cheeger and Gromoll

The decomposition theorem of Cheeger and Gromoll is a theorem from the mathematical field of differential geometry . It is of importance for the classification of Riemannian manifolds of nonnegative Ricci curvature .

sentence

Let be a complete Riemannian manifold of nonnegative Ricci curvature, so . If there is an embedded geodesic , then isometric to Ricci curvature nonnegative for a Riemann manifold , thus . ${\displaystyle M}$ ${\displaystyle Ric(M)\geq 0}$ ${\displaystyle \gamma \colon \mathbb {R} \to M}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} \times N}$ ${\displaystyle N}$ ${\displaystyle Ric(N)\geq 0}$

history

For surfaces , the theorem was proven by Stefan Cohn-Vossen in 1936 . In 1959, WA Toponogow proved the general theorem for manifolds of nonnegative sectional curvature . In 1971 Jeff Cheeger and Detlef Gromoll found the sentence in the above form with the weaker condition of nonnegative Ricci curvature .

The theorem was later proved for Lorentz manifolds whose Ricci curvature in the spatial direction is nonnegative.

literature

• E. Heintze , J.-H. Eschenburg : An elementary proof of the Cheeger-Gromoll splitting theorem , Ann. Glob. Anal. and Geom. 2: 141-151 (1984).

Individual evidence

1. ↑ S. Cohn-Vossen: Total curvature and geodetic lines on simply connected open complete areas. Матем. сб., 1 (43): 2 (1936), 139-164.
2. ^ VA Toponogov: Riemannian spaces containing straight lines. (Russian) Docl. Akad. Nauk SSSR 127 (1959): 977-979.
3. ^ Jeff Cheeger, Detlef Gromoll: The splitting theorem for manifolds of non-negative Ricci curvature. Journal of Differential Geometry 6 (1971/72), 119-128.
4. ↑ J.-H. Eschenburg: The splitting theorem for space-times with strong energy condition. J. Differential Geom. 27 (1988) no. 3, 477-491.
5. ^ Gregory Galloway : The Lorentzian splitting theorem without the completeness assumption. J. Differential Geom. 29 (1989), no. 2, 373-387.
6. ^ Richard PAC Newman : A proof of the splitting conjecture of S.-T. Yau. J. Differential Geom. 31 (1990) no. 1, 163-184.