On the Extrinsic Geometric Properties of Parabolic Surfaces and Topological Properties of Saddle Surfaces in Symmetric Spaces of Rank One

Borisenko, A. A.

doi:10.1070/sm1983v044n03abeh000974

Cited by 7 publications

(19 citation statements)

References 15 publications

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“…Corollary 1 Let N 2n 1 1 be a compact complex analytic submanifold and N n 2 2 be a (k, ε)-saddle submanifold in a complete connected Kählerian manifold M 2l . Suppose that for some q ≤ min{n 1 , 1 2 (n 2 −k)} and c > 0 the q-bisectional curvature satisfies the inequality H q (M)≥qc.…”

Section: Definitionmentioning

confidence: 99%

“…Borisenko [2], proved the vanishing of the intermediate homology groups of compact k-saddle submanifolds in a sphere S l (K) and projective spaces CP l (4K), HP l (4K) with Fubini metric of sectional curvature K ≤ K σ ≤ 4K).…”

mentioning

confidence: 98%

“…The definition of a k-saddle submanifold is equivalent to the existence of asymptotic subspaces of each normal of a submanifold. The last property is essential in studies of deformations of n-dimensional submanifolds of codimension p = n(n+1)/2, see discussion in [3].…”

mentioning

confidence: 98%

“…The k-saddle and k-parabolic (i.e., with the rank r(N) ≤ n − k) submanifolds were intensively studied by Borisenko et al [4] and Borisenko [3], etc. The definition of a k-saddle submanifold is equivalent to the existence of asymptotic subspaces of each normal of a submanifold.…”

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confidence: 99%

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On $$(k,\varepsilon)$$ -saddle submanifolds of Riemannian manifolds

Rovenski

2006

Geom Dedicata

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The paper continues our (in collaboration with A. Borisenko [J. Differential Geom. Appl. 20 p., to appear]) discovery of the new classes of (k, ε)-saddle, (k, ε)-parabolic, and (k, ε)-convex submanifolds ( ε ≥ 0). These are defined in terms of the eigenvalues of the 2nd fundamental forms of each unit normal of the submanifold, extending the notion of k-saddle, k-parabolic, k-convex submanifolds ( ε = 0). It follows that the definition of (k, ε)-saddle submanifolds is equivalent to the existence of ε-asymptotic subspaces in the tangent space. We prove non-embedding theorems of (k, ε)-saddle submanifolds, theorems about 1-connectedness and homology groups of these submanifolds in Riemannian spaces of positive (sectional or qth Ricci) curvature, in particular, spherical and projective spaces. We apply these results to submanifolds with 'small' normal curvature, k n ≤ ε, and for submanifolds with extrinsic curvature ≤ ε 2 (resp., non-positive) and small codimension, and include some illustrative examples. The results of the paper generalize theorems about totally geodesic, minimal and k-saddle submanifolds by Frankel; Borisenko, Rabelo and Tenenblat; Kenmotsu and Xia; Mendonça and Zhou.

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Section: Definitionmentioning

confidence: 99%

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confidence: 98%

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confidence: 98%

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confidence: 99%

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On $$(k,\varepsilon)$$ -saddle submanifolds of Riemannian manifolds

Rovenski

2006

Geom Dedicata

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“…Hence at least k eigenvalues of the (symmetric) Weingarten operator A ξ are non-negative. In this case, the homology groups (with integer coefficients) H n−1−k+s (∂K) = 0 for 0 < s < 2k − n − 1, see Lemma 16 in [12] and also Lemma 6(b) in [9].…”

Section: Complex or Quaternion) And G Be A Group Of All Either Translmentioning

confidence: 99%

Some extensions of the class of k-convex bodies

Golubyatnikov

Rovenski

2009

Sib Math J

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Abstract.1 We study relations of some classes of k-convex, k-visible bodies in Euclidean spaces. We introduce and study circular projections in normed linear spaces and classes of bodies related with families of such maps, in particular, k-circular convex and k-circular visible ones. Investigation of these bodies more general than k-convex and k-visible ones allows us to generalize some classical results of geometric tomography and find their new applications.

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