The soul at infinity in dimension 4

Walschap, Gerard

doi:10.1090/S0002-9939-1991-1049852-X

The soul at infinity in dimension $4$
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by Gerard Walschap

Proc. Amer. Math. Soc. 112 (1991), 563-567

DOI: https://doi.org/10.1090/S0002-9939-1991-1049852-X

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Abstract:

It is shown that $4$-dimensional open manifolds with nonnegative sectional curvature whose fibers are totally geodesic are metrically rigid. For arbitrary dimension, one also concludes that the curvature is in a sense maximal at the soul.

References

Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
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
S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Universitext, Springer-Verlag, Berlin, 1987. MR 909697, DOI 10.1007/978-3-642-97026-9
Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865
V. A. Šarafutdinov, The Pogorelov-Klingenberg theorem for manifolds that are homeomorphic to $\textbf {R}^{n}$, Sibirsk. Mat. Ž. 18 (1977), no. 4, 915–925, 958 (Russian). MR 0487896
Gerard Walschap, Nonnegatively curved manifolds with souls of codimension $2$, J. Differential Geom. 27 (1988), no. 3, 525–537. MR 940117
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