Geodesic
On the extrinsic geometric properties of parabolic surfaces and topological properties of saddle surfaces in symmetric spaces of rank one
Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 401-415 
A. A. Borisenko. On the extrinsic geometric properties of parabolic surfaces and topological properties of saddle surfaces in symmetric spaces of rank one. Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 401-415. http://geodesic.mathdoc.fr/item/SM_1983_44_3_a9/
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     title = {On the extrinsic geometric properties of parabolic surfaces and topological properties of saddle surfaces in symmetric spaces of rank one},
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     volume = {44},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1983_44_3_a9/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

This paper investigates the metric structure of compact $k$-parabolic surfaces and topological properties of $k$-saddle surfaces in the sense of Shefel' in symmetric spaces of rank one, namely, spherical space $S^n$, complex projective space $CP^n$, and quaternion projective space $QP^n$. It turns out that $k$-parabolic surfaces for large $k$ are totally geodesic spheres $S^l$ in $S^n$, totally geodesic complex projective spaces $CP^l$ in $CP^n$, and totally geodesic quaternion projective spaces $QP^l$ in $QP^n$. It follows that surfaces of nonpositive extrinsic $q$-dimensional curvature, under a natural restriction on the codimension of the embedding, are totally geodesic surfaces in $S^n$, $CP^n$ and $QP^n$. Saddle surfaces for small $k$ have restrictions on the homology and cohomology groups. Since surfaces of nonpositive $q$-dimensional extrinsic curvature for small codimension of the embedding are $k$-saddle surfaces, they also have degeneracies in the homology and cohomology groups. Bibliography: 27 titles.

[1] Chern S. S., Kuiper N. H., “Some theorems on the isometric imbedding of compact Riemann manifolds in Euclidean space”, Ann. Math., 56 (1952), 422–430 | DOI | MR

[2] Chen B.-Y., Ogiue K., “On totally real submanifolds”, Trans. Amer. Math. Soc., 193 (1974), 257–266 | DOI | MR | Zbl

[3] Chen B.-Y., “Totally umbilical submanifolds of quaternon-space-forms”, J. Austra Math. Soc., 26 (1978), 154–162 | DOI | MR | Zbl

[4] Kobayascht S., Nomizu K., Foundation of Differential Geometry, 2, New York, 1969

[5] Bishop R. L., Goldberg S. I., “On the second cohomology group of a Kahlier manifold of positive curvature”, Proc. Amer. Math. Soc., 16:1 (1965), 119–122 | DOI | MR | Zbl

[6] Borisenko A. A., “Polnye $l$-mernye poverkhnosti nepolozhitelnoi vneshnei krivizny v rimanovom prostranstve”, Matem. sb., 104(146) (1977), 559–577 | MR

[7] Borisenko A. A., “O poverkhnostyakh nepolozhitelnoi vneshnei krivizny v prostranstvakh postoyannoi krivizny”, Matem. sb., 114(156) (1981), 339–354 | MR | Zbl

[8] Shefel S. Z., “O dvukh klassakh $k$-mernykh poverkhnostei v $n$-mernom evklidovom prostranstve”, Sib. matem. zh., X:2 (1969), 459–467

[9] Glazyrin V. V., “Topologicheskie i metricheskie svoistva $k$-sedlovykh poverkhnostei”, DAN SSSR, 233:6 (1977), 1028–1030 | MR | Zbl

[10] Milnor Dzh., Teoriya Morsa, Mir, M., 1965 | MR

[11] Borisenko A. A., “O kharakteristicheskikh klassakh Pontryagina kompaktnykh poverkhnostei v sfericheskom prostranstve”, Ukr. geom. sb., 1980, no. 23, 24–30 | MR | Zbl

[12] Larsen M. E., “On the topology of complex projective manifolds”, Invent. math., 19:3 (1973), 251–260 | DOI | MR | Zbl

[13] Maltz R., “The nullity spaces of curvature-like tensors”, J. Differ. geom., 1972, no. 3–4, 519–525 | MR

[14] Neill B. O., “The fundamental equations of a submersion”, Michig. Math. J., 13 (1966), 459–469 | DOI | MR | Zbl

[15] Escobales R. H., “Riemannian submersions from complex projective space”, J. Differ. geom., 13:1 (1978), 93–107 | MR | Zbl

[16] Escobales R. H., “Riemannian submersions with totally geodesic fibers”, J. Differ. geom., 10 (1975), 253–276 | MR | Zbl

[17] Borisenko A. A., “O stroenii $l$-mernykh poverkhnostei s vyrozhdennoi kvadratichnoi formoi”, Ukr. geom. sb., 1972, no. 13, 18–27 | MR

[18] Frankel T., “Manifolds with positive curvature”, Pacific. J. Math., 11 (1961), 165–174 | MR | Zbl

[19] Goldberg S. I., Kobayashi S., “On holomorphic bisectional curvature”, J. Differ. geom., 1 (1967), 225–233 | MR | Zbl

[20] Borisenko A. A., “O polnykh poverkhnostyakh v prostranstvakh postoyannoi krivizny”, Ukr. geom. sb., 1974, no. 15, 8–15 | MR | Zbl

[21] Cheng S. Y., “Eigenvalue comparison theorems and its geometrical applications”, Math. Z., 143 (1975), 289–297 | DOI | MR | Zbl

[22] Moore J. D., “Submanifolds of constant positive curvature”, Duke Math. J., 44:2 (1978), 449–484 | DOI | MR

[23] Spener E., Algebraicheskaya topologiya, Mir, M., 1971 | MR

[24] Ogiue K., “Differential geometry of Kahler submanifolds”, Adv. Math., 13 (1974), 73–114 | DOI | MR | Zbl

[25] Milnor Dzh., Teorema o $h$-kobordizme, Mir, M., 1969 | MR

[26] Gromol D., Klingenberg V., Meier V., Rimanova geometriya v tselom, Mir, M., 1971