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Cycle-parallel real hypersurfaces of quaternionic projective space

1993; University of Tsukuba; Volume: 17; Issue: 1 Linguagem: Inglês

10.21099/tkbjm/1496162139

2423-821X

Juan de Dios Pérez,

Advanced Topics in Algebra

We classify cyclic-parallel real hypersurfaces of quater- nionic proiective space.1. Introduction.Let $M$ be a connected real hypersurface of a quaternionic projective space $QP^{m},$ $m\geqq 2$ , endowed with the metric of constant quaternionic sectional curvature 4. If $\zeta$ denotes the unit local normal vector field and $\{J_{1}, J_{2}, J_{3}\}$ a local basis of the quaternionic structure of $QP^{m}$ (see [2]), then $U_{i}=-J_{i}\zeta,$ $i=1,2,3$, are vector fields tangent to $M$ .It is known, [4], that there do not exist parallel real hypersurfaces of $QP^{m},$ $m\geqq 2$ .A real hypersurface of $QP^{m}$ is called cyclic-parallel if it satisfies (1.1) $\sigma(g((\nabla_{X}A)Y, Z))=0$ for any $X,$ $Y,$ $Z$ tangent to $M$ , where $A$ denotes the Weingarten endomorphism of $M$ and $\sigma$ the cyclic sum.Our purpose is to classify such real hypersurfaces by mean of the following THEOREM.A real hypersurface $M$ of $QP^{m},$ $m\geqq 2$ , is cyclic-parallel if and only if it is congruent to an open subset of a tube of radius $r,$ $0<r<\Pi/2$, over $QP^{k},$ $k\in\{0, \cdots\prime m-1\}$ .In the Theorem, $QP^{k}$ is considered canonically and totally geodesically embedded in $QP^{m}$ . Preliminaries.Let $X$ be a vector field tangent to $M$ .We write $J_{i}X=\phi_{i}X+f_{i}(X)\zeta,$ $i=1$ , 2, 3, where $\phi_{i}X$ denotes the tangent component of $J_{i}X$ and $f_{i}(X)=g(X, U_{i})$ .From this, see [3], we have