On manifolds with nonnegative curvature on totally isotropic 2-planes

Seaman, Walter

doi:10.1090/S0002-9947-1993-1123458-2

On manifolds with nonnegative curvature on totally isotropic 2-planes
HTML articles powered by AMS MathViewer

by Walter Seaman

Trans. Amer. Math. Soc. 338 (1993), 843-855

DOI: https://doi.org/10.1090/S0002-9947-1993-1123458-2

PDF | Request permission

Abstract:

We prove that a compact orientable $2n$-dimensional Riemannian manifold, with second Betti number nonzero, nonnegative curvature on totally isotropic $2$-planes, and satisfying a positivity-type condition at one point, is necessarily Kähler, with second Betti number $1$. Using the methods of Siu and Yau, we prove that if the positivity condition is satisfied at every point, then the manifold is biholomorphic to complex projective space.

References

Shigetoshi Bando, On the classification of three-dimensional compact Kaehler manifolds of nonnegative bisectional curvature, J. Differential Geom. 19 (1984), no. 2, 283–297. MR 755227
James Eells and Luc Lemaire, Selected topics in harmonic maps, CBMS Regional Conference Series in Mathematics, vol. 50, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 703510, DOI 10.1090/cbms/050
Akito Futaki, On compact Kähler manifolds with semipositive bisectional curvature, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 1, 111–125. MR 617868
Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121–138 (French). MR 87176, DOI 10.2307/2372388
Samuel I. Goldberg and Shoshichi Kobayashi, Holomorphic bisectional curvature, J. Differential Geometry 1 (1967), 225–233. MR 227901
Shoshichi Kobayashi, On compact Kähler manifolds with positive definite Ricci tensor, Ann. of Math. (2) 74 (1961), 570–574. MR 133086, DOI 10.2307/1970298

Foundations of differential geometry

Shoshichi Kobayashi and Takushiro Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31–47. MR 316745, DOI 10.1215/kjm/1250523432
Mario J. Micallef and John Douglas Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199–227. MR 924677, DOI 10.2307/1971420
J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI 10.2307/1971131
Walter Seaman, Orthogonally pinched curvature tensors and applications, Math. Scand. 69 (1991), no. 1, 5–14. MR 1143470, DOI 10.7146/math.scand.a-12365
Yum Tong Siu, Curvature characterization of hyperquadrics, Duke Math. J. 47 (1980), no. 3, 641–654. MR 587172
Yum Tong Siu and Shing Tung Yau, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), no. 2, 189–204. MR 577360, DOI 10.1007/BF01390043
Hung Hsi Wu, The Bochner technique in differential geometry, Math. Rep. 3 (1988), no. 2, i–xii and 289–538. MR 1079031
Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. MR 480350, DOI 10.1002/cpa.3160310304