Focal sets and real hypersurfaces in complex projective space
Authors:
Thomas E. Cecil and Patrick J. Ryan
Journal:
Trans. Amer. Math. Soc. 269 (1982), 481499
MSC:
Primary 53C40; Secondary 53C15
MathSciNet review:
637703
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Abstract: Let be a real submanifold of , and let denote the complex structure. We begin by finding a formula for the location of the focal points of in terms of its second fundamental form. This takes a particularly tractable form when is a complex submanifold or a real hypersurface on which is a principal vector for each unit normal to . The rank of the focal map onto a sheet of the focal set of is also computed in terms of the second fundamental form. In the case of a real hypersurface on which is principal with corresponding principal curvature , if the map onto a sheet of the focal set corresponding to has constant rank, then that sheet is a complex submanifold over which is a tube of constant radius (Theorem 1). The other sheets of the focal set of such a hypersurface are given a real manifold structure in Theorem 2. These results are then employed as major tools in obtaining two classifications of real hypersurfaces in . First, there are no totally umbilic real hypersurfaces in , but we show: Theorem 3. Let be a connected real hypersurface in , , with at most two distinct principal curvatures at each point. Then is an open subset of a geodesic hypersphere. Secondly, we show that there are no Einstein real hypersurfaces in and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the Ricci tensor in Theorem 4.
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 [1]
 W. Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press, New York, 1975. MR 0426007 (54:13956)
 [2]
 T. Cecil, Taut immersions of noncompact surfaces into a Euclidean space, J. Differential Geom. 11 (1976), 451459. MR 0438360 (55:11275)
 [3]
 , Geometric applications of critical point theory to submanifolds of complex projective space, Nagoya Math. J. 55 (1974), 531. MR 0350672 (50:3164)
 [4]
 T. Cecil and P. Ryan, Focal sets of submanifolds, Pacific J. Math. 78 (1978), 2739. MR 513280 (80a:53059)
 [5]
 , Focal sets, taut embeddings and the cyclides of Dupin, Math. Ann. 236 (1978), 177190. MR 503449 (80a:53003)
 [6]
 B.Y. Chen, Extrinsic spheres in Riemannian manifolds, Houston J. Math. 5 (1979), 319324. MR 559971 (81h:53053)
 [7]
 B.Y. Chen and K. Ogiue, Two theorems on Kaehler manifolds, Michigan Math. J. 21 (1974), 225229. MR 0367884 (51:4126)
 [8]
 A. Fialkow, Hypersurfaces of a space of constant curvature, Ann. of Math. (2) 39 (1938), 762785. MR 1503435
 [9]
 S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vols. 1, 2, Interscience, New York, 1963, 1969. MR 0152974 (27:2945)
 [10]
 M. Kon, PseudoEinstein real hypersurfaces in complex space forms, J. Differential Geom. 14 (1979), 339354. MR 594705 (81k:53050)
 [11]
 Y. Maeda, On real hypersurfaces of a complex projective space, J. Math. Soc. Japan 28 (1976), 529540. MR 0407772 (53:11543)
 [12]
 K. Nomizu, Elie Carton's work on isoparametric families of hypersurfaces, Proc. Sympos. Pure Math., vol. 27, Part I, Amer. Math. Soc., Providence, R. I., 1974, pp. 191200. MR 0423260 (54:11240)
 [13]
 K. Nomizu and B. Smyth, Differential geometry of complex hypersurfaces. II, J. Math. Soc. Japan 20 (1968), 498521. MR 0230264 (37:5827)
 [14]
 B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459469. MR 0200865 (34:751)
 [15]
 R. Palais, A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc. No. 22(1957). MR 0121424 (22:12162)
 [16]
 H. Reckziegel, On the eigenvalues of the shape operator of an isometric immersion into a space of constant curvature, Math. Ann. 243 (1979), 7182. MR 543096 (80h:53057)
 [17]
 P. Ryan, Hypersurfaces with parallel Ricci tensor, Osaka J. Math. 8 (1971), 251259. MR 0296859 (45:5918)
 [18]
 , Homogeneity and some curvature conditions for hypersurfaces, Tôhoku Math. J. 21 (1969), 363388. MR 0253243 (40:6458)
 [19]
 B. Smyth, Differential geometry of complex hypersurfaces, Ann. of Math. (2) 85 (1967), 246266. MR 0206881 (34:6697)
 [20]
 R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975), 4353. MR 0355906 (50:8380)
 [21]
 , Real hypersurfaces in a complex projective space with constant principal curvatures, II, J. Math. Soc. Japan 27 (1975), 507516. MR 0400120 (53:3955)
 [22]
 Y. Tashiro and S. Tachibana, On Fubiman and Fubinian manifolds, Kōdai Math. Sem. Rep. 15 (1963), 176183. MR 0157336 (28:571)
 [23]
 T. Thomas, Extract from a letter by E. Carian concerning my note: On closed spaces of constant mean curvature, Amer. J. Math. 59 (1937), 793794. MR 1507282
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 A. Weinstein, Distance spheres in complex projective spaces, Proc. Amer. Math. Soc. 39 (1973), 649650. MR 0315631 (47:4180)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206377033
PII:
S 00029947(1982)06377033
Keywords:
Focal sets,
real hypersurfaces in complex projective space,
umbilic hypersurfaces,
Einstein hypersurfaces
Article copyright:
© Copyright 1982
American Mathematical Society
