Submanifolds in differentiable manifolds endowed with differential-geometric structures VI.CR-submanifolds in a manifold of almost-contact structure

In this paper, we introduce a Riemannian metric [Formula: see text] and a family of framed f-structures on the slit tangent bundle [Formula: see text] of a Finsler manifold Fn = (M, F). Then we prove that there exists an almost contact structure on the tangent bundle, when this structure is restricted to the Finslerian indicatrix. We show that this structure is Sasakian if and only if Fn is of positive constant curvature 1. Finally, we prove that (i) Fn is a locally flat Riemannian manifold if and only if [Formula: see text], (ii) the Jacobi operator [Formula: see text] is zero or commuting if and only if (M, F) have the zero flag curvature.

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Submanifolds in differentiable manifolds endowed with differential-geometric structures VI.CR-submanifolds in a manifold of almost-complex structure

Journal of Soviet Mathematics ◽

10.1007/bf01098652 ◽

1989 ◽

Vol 44 (2) ◽

pp. 123-152

Author(s):

N. M. Ostianu

Keyword(s):

Complex Structure ◽

Almost Complex Structure ◽

Geometric Structures ◽

Differentiable Manifolds ◽

Almost Complex

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Submanifolds of differentiable manifolds endowed with differential-geometric structures. IV. Submanifolds of codimension 1 of manifolds with an almost-complex structure

Journal of Soviet Mathematics ◽

10.1007/bf02105051 ◽

1985 ◽

Vol 29 (5) ◽

pp. 1592-1616

Author(s):

T. N. Balazyuk ◽

N. M. Ostianu

Keyword(s):

Complex Structure ◽

Almost Complex Structure ◽

Geometric Structures ◽

Differentiable Manifolds ◽

Almost Complex

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Normal Variations of Invariant Hypersurfaces of Framed Manifolds

Canadian Mathematical Bulletin ◽

10.4153/cmb-1972-090-x ◽

1972 ◽

Vol 15 (4) ◽

pp. 513-521

Author(s):

Samuel I. Goldberg

Keyword(s):

Complex Manifold ◽

Contact Structure ◽

Complex Structure ◽

Almost Complex Structure ◽

Marked Contrast ◽

Almost Complex Manifold ◽

Almost Contact Structure ◽

Almost Complex ◽

Differentiable Vector ◽

Real Codimension

A hypersurface of a globally framed f-manifold (briefly, a framed manifold), does not in general possess a framed structure as one may see by considering the 4-sphere S4 in R5 or S5. For, a hypersurface so endowed carries an almost complex structure, or else, it admits a nonsingular differentiable vector field. Since an almost complex manifold may be considered as being globally framed, with no complementary frames, this situation is in marked contrast with the well known fact that a hypersurface (real codimension 1) of an almost complex manifold admits a framed structure, more specifically, an almost contact structure.

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