On Totally Real Submanifolds in a 6-Sphere

1990 ◽  
Vol 33 (2) ◽  
pp. 162-166
Author(s):  
M. A. Bashir
Keyword(s):  
Complex Structure ◽  
Dimensional Sphere ◽  
Almost Complex Structure ◽  
Real Submanifolds ◽  
Cayley Algebra ◽  
Almost Complex ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Totally Real ◽  
Kaehlerian Manifold

AbstractThe 6-dimensional sphere S6 has an almost complex structure induced by properties of Cayley algebra. With respect to this structure S6 is a nearly Kaehlerian manifold. We investigate 2-dimensional totally real submanifolds in S6. We prove that a 2-dimensional totally real submanifold in S6 is flat.

scholarly journals TOTAL REALITY OF CONORMAL BUNDLES OF HYPERSURFACES IN ALMOST COMPLEX MANIFOLDS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1255-1262 ◽  
Author(s):  
ANDREA SPIRO
Keyword(s):  
Complex Structure ◽  
Almost Complex Structure ◽  
Almost Complex Manifolds ◽  
Almost Complex Manifold ◽  
Conormal Bundle ◽  
Almost Complex ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Totally Real ◽  
Pseudoconvex Hypersurface

A generalization to the almost complex setting of a well-known result by Webster is given. Namely, we prove that if Γ is a strongly pseudoconvex hypersurface in an almost complex manifold (M, J), then the conormal bundle of Γ is a totally real submanifold of (T* M, 𝕁), where 𝕁 is the lifted almost complex structure on T* M defined by Ishihara and Yano.

Smooth Boundary Values Along Totally Real Submanifolds

1984 ◽  
Vol 36 (2) ◽  
pp. 240-248 ◽  
Author(s):  
Edgar Lee Stout
Keyword(s):  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Regularity Result ◽  
Boundary Values ◽  
Real Submanifolds ◽  
Strongly Pseudoconvex Domain ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Class C ◽  
Totally Real

The main result of this paper is the following regularity result:THEOREM. Let D ⊂ CNbe a bounded, strongly pseudoconvex domain with bD of class Ck, k ≧ 3. Let Σ ⊂ bD be an N-dimensional totally real submanifold, and let f ∊ A(D) satisfy |f| = 1 on Σ, |f| < 1 on. If Σ is of class Cr, 3 ≦ r < k, then the restriction fΣ = f|Σ of f to Σ is of class Cr − 0, and if Σ is of class Ck, then fΣ is of class Ck − 1.Here, of course, A(D) denotes the usual space of functions continuous on , holomorphic on D, and we shall denote by Ak(D), k = 1, 2, …, the space of functions holomorphic on D whose derivatives or order k lie in A(D).

scholarly journals TOTALLY REAL SURFACES IN $S^6$

2021 ◽  
Vol 23 (1) ◽  
pp. 11-14
Author(s):  
SHARIEF DESHMUKH
Keyword(s):  
Gaussian Curvature ◽  
Complex Structure ◽  
Second Fundamental Form ◽  
Real Surface ◽  
Almost Complex Structure ◽  
Constant Gaussian Curvature ◽  
Totally Geodesic ◽  
The Second Fundamental Form ◽  
Almost Complex ◽  
Totally Real

The normal bundle $\bar \nu$ of a totally real surface $M$ in $S^6$ splits as $\bar\nu= JTM\oplus \bar\mu$ where $TM$ is the tangent bundle of $M$ and  $\bar\mu$ is sub­bundle of $\bar\nu$ which is invariant under the almost complex structure $J$. We study the totally real surfaces M of constant Gaussian curvature K for which the second fundamental form $h(x, y) \in JTM$, and we show that $K = 1$ (that is, $M$ is totally geodesic).

XX.—Differential Geometry on Hypersurfaces in a Cayley Space

1964 ◽  
Vol 66 (4) ◽  
pp. 216-231 ◽  
Author(s):  
K. Yano ◽  
T. Sumitomo
Keyword(s):  
Complex Structure ◽  
Hermitian Structure ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Almost Complex Structure ◽  
Principal Curvatures ◽  
Totally Umbilical ◽  
Almost Complex ◽  
Necessary And Sufficient

A seven-dimensional Euclidean space considered as the space of purely imaginary Cayley numbers is called a Cayley space. The six-dimensional sphere in a Cayley space admits an almost complex structure which is not integrable. Moreover the algebraic properties of the imaginary Cayley numbers induce an almost complex structure on any oriented differentiable hypersurface in the Cayley space. The Riemannian metric induced on the hypersurface from the metric of the Cayley space is Hermitian with respect to the almost complex structure.It is proved that the induced Hermitian structure of an oriented hypersurface in the Cayley space is almost Kaehlerian if and only if it is Kaehlerian, that a necessary and sufficient condition for a hypersurface in a Cayley space to be an almost Tachibana space is that the hypersurface be totally umbilical, and that a totally umbilical hypersurface in a Cayley space admits a complex structure when and only when it is totally geodesic.For a hypersurface in the Cayley space with the induced Hermitian structure which is an *O-space it is proved that all the principal curvatures of the hypersurface are constant, and from this is deduced a classification of such *O-spaces.

scholarly journals Remarks on a totally real submanifold

1975 ◽  
Vol 51 (1) ◽  
pp. 5-6 ◽  
Author(s):  
Seiichi Yamaguchi ◽  
Toshihiko Ikawa
Keyword(s):  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Totally Real

The almost complex structure on 𝕊6 and related Schrödinger flows

2018 ◽  
Vol 29 (14) ◽  
pp. 1850099 ◽  
Author(s):  
Qing Ding ◽  
Shiping Zhong
Keyword(s):  
Complex Structure ◽  
Type System ◽  
Text Structure ◽  
Almost Complex Structure ◽  
Nls Equation ◽  
Geometric Properties ◽  
Nonlinear Schrödinger ◽  
Almost Complex ◽  
Complex Functions

In this paper, by using the [Formula: see text]-structure on Im[Formula: see text] from the octonions [Formula: see text], the [Formula: see text]-binormal motion of curves [Formula: see text] in [Formula: see text] associated to the almost complex structure on [Formula: see text] is studied. The motion is proved to be equivalent to Schrödinger flows from [Formula: see text] to [Formula: see text], and also to a nonlinear Schrödinger-type system (NLSS) in three unknown complex functions that generalizes the famous correspondence between the binormal motion of curves in [Formula: see text] and the focusing nonlinear Schrödinger (NLS) equation. Some related geometric properties of the surface [Formula: see text] in Im[Formula: see text] swept by [Formula: see text] are determined.

scholarly journals BRANCHED SHADOWS AND COMPLEX STRUCTURES ON 4-MANIFOLDS

2008 ◽  
Vol 17 (11) ◽  
pp. 1429-1454 ◽  
Author(s):  
FRANCESCO COSTANTINO
Keyword(s):  
Complex Structure ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
Homotopy Classes ◽  
Almost Complex Structures ◽  
Almost Complex

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and of Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with Spinc-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-manifolds and prove that each almost complex structure on a 4-dimensional handlebody is homotopic to a complex one.

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