scholarly journals On the differentiability of a distance function

2008 ◽  
Vol 83 (97) ◽  
pp. 65-69
Author(s):  
Kwang-Soon Park
Keyword(s):  
Distance Function ◽  
Normal Bundle ◽  
Totally Geodesic ◽  
Complex Submanifold ◽  
Minimal Geodesic ◽  
Simply Connected ◽  
Unit Normal

Let M be a simply connected complete K?hler manifold and N a closed complete totally geodesic complex submanifold of M such that every minimal geodesic in N is minimal in M. Let U? be the unit normal bundle of N in M. We prove that if a distance function ? is differentiable at v ? U?, then ? is also differentiable at -v.

Normal and osculating maps for submanifolds of RN

1990 ◽  
Vol 114 (1-2) ◽  
pp. 39-55 ◽  
Author(s):  
I. Cattaneo Gasparini ◽  
G. Romani
Keyword(s):  
Isometric Immersion ◽  
Normal Bundle ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Gauss Map ◽  
Necessary And Sufficient Conditions ◽  
Totally Geodesic ◽  
Precise Formulation ◽  
Parallel Case ◽  
Necessary And Sufficient

SynopsisLet Mn be a manifold supposed “nicely curved” isometrically immersed in ℝn+p. Starting from a generalised Gauss map associated to the splitting of the normal bundle defined from the values of the fundamental forms of M of order k (k ≧ 0), we give necessary and sufficient conditions for the map to be totally geodesic and harmonic . For k = 0 is the classical Gauss map and our formula reduces to Ruh–Vilm's formula with a more precise formulation due to the consideration of the splitting of the normal bundle.We also give necessary conditions for M, supposed complete, to admit an isometric immersion with . This theorem generalises a theorem of Vilms on the manifolds with second fundamental forms parallel (case k = 0). The result is interesting as the class of manifolds satisfying the condition is larger than the class of manifolds satisfying .

scholarly journals A BARTH–LEFSCHETZ THEOREM FOR SUBMANIFOLDS OF A PRODUCT OF PROJECTIVE SPACES

2009 ◽  
Vol 20 (01) ◽  
pp. 77-96
Author(s):  
LUCIAN BĂDESCU ◽  
FLAVIA REPETTO
Keyword(s):  
Normal Bundle ◽  
Projective Spaces ◽  
Restriction Map ◽  
Vanishing Theorem ◽  
Type Result ◽  
Picard Groups ◽  
Lefschetz Type ◽  
Simply Connected ◽  
Join Construction ◽  
Torsion Free

Let X be a complex submanifold of dimension d of ℙm × ℙn (m ≥ n ≥ 2) and denote by α: Pic(ℙm × ℙn) → Pic(X) the restriction map of Picard groups, by NX|ℙm × ℙn the normal bundle of X in ℙm × ℙn. Set t := max{dim π1(X), dim π2(X)}, where π1 and π2 are the two projections of ℙm × ℙn. We prove a Barth–Lefschetz type result as follows: Theorem. If [Formula: see text] then X is algebraically simply connected, the map α is injective and Coker(α) is torsion-free. Moreover α is an isomorphism if [Formula: see text], or if [Formula: see text] and NX|ℙm×ℙn is decomposable. These bounds are optimal. The main technical ingredients in the proof are: the Kodaira–Le Potier vanishing theorem in the generalized form of Sommese ([18, 19]), the join construction and an algebraization result of Faltings concerning small codimensional subvarieties in ℙN (see [9]).

scholarly journals Chen's problem on mixed foliate CR-submanifolds

1989 ◽  
Vol 40 (1) ◽  
pp. 157-160 ◽  
Author(s):  
Mohammed Ali Bashir
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Complex Space Form ◽  
Complex Submanifold ◽  
Hyperbolic Complex ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Simply Connected ◽  
Totally Real ◽  
Cr Submanifolds

We prove that the simply connected compact mixed foliate CR-submanifold in a hyperbolic complex space form is either a complex submanifold or a totally real submanifold. This is the problem posed by Chen.

scholarly journals Foliation by G-orbits

2014 ◽  
Vol 33 (2) ◽  
pp. 79-87
Author(s):  
Jose Rosales-Ortega
Keyword(s):  
Riemannian Manifold ◽  
Lie Group ◽  
Normal Bundle ◽  
Semisimple Lie Group ◽  
Totally Geodesic

We study the properties of the normal bundle defined by the bundle of the G-orbits of the action of a semisimple Lie group G on a pseudo Riemannian manifold M, as a consequence we obtain that the foliation induced by the normal bundle is integrable and totally geodesic.

On Kaehler Immersions

1972 ◽  
Vol 24 (6) ◽  
pp. 1178-1182 ◽  
Author(s):  
Koichi Ogiue
Keyword(s):  
Projective Space ◽  
Sectional Curvature ◽  
Complex Projective Space ◽  
Holomorphic Sectional Curvature ◽  
Kaehler Manifold ◽  
Totally Geodesic ◽  
Complex Submanifold ◽  
Complex Projective

Let be an (n + p)-dimensional Kaehler manifold of constant holomorphic sectional curvature . B. O'Neill [3] proved the following result.PROPOSITION A. Let M be an n-dimensional complex submanifold immersed in . If and if the holomorphic sectional curvature of M with respect to the induced Kaehler metric is constant, then M is totally geodesic.He also gave the following example: There is a Kaehler imbedding of an w-dimensional complex projective space of constant holomorphic sectional curvature ½ into an -dimensional complex projective space of constant holomorphic sectional curvature 1. This shows that Proposition A is the best possible.

scholarly journals REAL HYPERSURFACES WITH ISOMETRIC REEB FLOW IN COMPLEX QUADRICS

2013 ◽  
Vol 24 (07) ◽  
pp. 1350050 ◽  
Author(s):  
JURGEN BERNDT ◽  
YOUNG JIN SUH
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
Real Hypersurface ◽  
Kähler Manifolds ◽  
Real Hypersurfaces ◽  
Kahler Manifolds ◽  
Totally Geodesic ◽  
Open Part ◽  
Complex Submanifold ◽  
Complex Projective

We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Qm = SOm+2/SOmSO2, m ≥ 3. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space ℂPk which is embedded canonically in Q2k as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q2k+1, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow.

scholarly journals An integral formula for hypersurfaces in space forms

1995 ◽  
Vol 37 (3) ◽  
pp. 337-341 ◽  
Author(s):  
Theodoros Vlachos
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Distance Function ◽  
Integral Formula ◽  
Constant Sectional Curvature ◽  
Position Vector ◽  
Space Forms ◽  
Image Position ◽  
Simply Connected

Let be an n+ 1-dimensional, complete simply connected Riemannian manifold of constant sectional curvature c and We consider the function r(·) = d(·, P0) where d stands for the distance function in and we denote by grad r the gradient of The position vector (see [1]) with origin P0 is defined as where ϕ(r)equalsr, if c = 0, c< 0 or c <0 respectively.

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