A splitting theorem for complete manifolds with nonnegative curvature operator

Author:
Maria Helena Noronha

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

MSC:
Primary 53C20

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

MathSciNet review:
933519

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Abstract | References | Similar Articles | Additional Information

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

**[CE]**J. Cheeger and D. Ebin,*Comparison theorems in Riemannian geometry*, North-Holland, Amsterdam, 1975. MR**0458335 (56:16538)****[CG]**J. Cheeger and D. Gromoll,*On the structure of complete manifolds of nonnegative curvature*, Ann. of Math. (2)**96**(1972), 413-443. MR**0309010 (46:8121)****[CY]**B. Chow and D. Yang,*A classification of compact Riemannian manifolds with nonnegative curvature operator*, preprint.**[GH]**S. Gallot and D. Meyer,*Opérateur de courbure et Laplacien des formes différentielles d'une variété Riemanniene*, J. Math. Pures Appl.**54**(1975), 285-304. MR**0454884 (56:13128)****[H]**R. Hermann,*On the differential geometry of foliations*, Ann. of Math. (2)**72**(1960), 445-457. MR**0142130 (25:5523)****[KW]**N. Kleinjohann and R. Walter,*Nonnegativity of the curvature operator and isotropy for isometric immersions*, Math. Z.**181**(1982), 129-142. MR**671719 (84k:53053)****[O]**B. O'Neill,*The fundamental equations of a submersion*, Michigan Math. J.**13**(1966), 459-469. MR**0200865 (34:751)****[S]**M. Strake,*A splitting theorem for open nonnegatively curved manifolds*, (to appear in Manuscripta Math.) MR**949821 (89g:53066)****[T]**V. A. Toponogov,*Spaces with straight lines*, Amer. Math. Soc. Transl.**37**(1964), 287-290.**[Wa]**G. Walschap,*A splitting theorem for**-dimensional manifolds of nonnegative curvature*, Proc. Amer. Math. Soc. (to appear) MR**958080 (89g:53068)****[We]**A. Weinstein,*Positive curved**-manifolds in*, J. Differential Geom.**4**(1970), 1-4. MR**0264562 (41:9154)****[Y]**J.-W. Yim,*Distance nonincreasing retraction on a complete open manifold of nonnegative curvature*, Ann. Global Anal. Geom. (to appear) MR**982765 (90a:53049)****[Y]**-,*Space of souls in a complete open manifold of nonnegative curvature*, preprint.

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Additional Information

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

Keywords:
Nonnegative curvature,
soul

Article copyright:
© Copyright 1989
American Mathematical Society