A splitting theorem for complete manifolds with nonnegative curvature operator

Noronha, Maria Helena

doi:10.1090/S0002-9939-1989-0933519-0

A splitting theorem for complete manifolds with nonnegative curvature operator
HTML articles powered by AMS MathViewer

by Maria Helena Noronha

Proc. Amer. Math. Soc. 105 (1989), 979-985

DOI: https://doi.org/10.1090/S0002-9939-1989-0933519-0

PDF | Request permission

Abstract:

It is shown that any complete, noncompact, simply connected Riemannian manifold with nonnegative curvature operator is isometric to the product of its compact soul (in the sense of Cheeger-Gromoll) and a complete manifold diffeomorphic to a Euclidean space

References

Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
Jeff Cheeger and Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443. MR 309010, DOI 10.2307/1970819

A classification of compact Riemannian manifolds with nonnegative curvature operator

S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. (9) 54 (1975), no. 3, 259–284 (French). MR 454884
Robert Hermann, On the differential geometry of foliations, Ann. of Math. (2) 72 (1960), 445–457. MR 142130, DOI 10.2307/1970226
Norbert Kleinjohann and Rolf Walter, Nonnegativity of the curvature operator and isotropy for isometric immersions, Math. Z. 181 (1982), no. 1, 129–142. MR 671719, DOI 10.1007/BF01214986
Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865
Martin Strake, A splitting theorem for open nonnegatively curved manifolds, Manuscripta Math. 61 (1988), no. 3, 315–325. MR 949821, DOI 10.1007/BF01258442

Spaces with straight lines

37

Gerard Walschap, A splitting theorem for $4$-dimensional manifolds of nonnegative curvature, Proc. Amer. Math. Soc. 104 (1988), no. 1, 265–268. MR 958080, DOI 10.1090/S0002-9939-1988-0958080-5
Alan Weinstein, Positively curved $n$-manifolds in $R^{n+2}$, J. Differential Geometry 4 (1970), 1–4. MR 264562
Jin-Whan Yim, Distance nonincreasing retraction on a complete open manifold of nonnegative sectional curvature, Ann. Global Anal. Geom. 6 (1988), no. 2, 191–206. MR 982765, DOI 10.1007/BF00133039

Space of souls in a complete open manifold of nonnegative curvature