§ 0. Introduction.A Ci?-structure on an odd dimensional differentiate manifold is a pair (3), J) of a 1-codimensional subbundle 3) of the tangent bundle and a complex structure / on 3) with certain integrability condition. In this paper we shall consider almost contact structures belonging to a Ci?-structure.Ci?-structures are recently developed by Burns-Shneider [3], Burns-Diederich-Shneider [2], Chern-Moser [4], Tanaka [10], Webster [11] [12] and so on. In particular, Tanaka [10] has treated almost contact structures with certain conditions belonging to a Ci?-structure and found canonical connections associated with them. Ishihara [5] has also considered almost contact structures in the Ci?-category and studied pseudo-conformal mappings. Our standpoint is similar to [5]. Our main purpose is to give a change of canonical connections associated with almost contact structures belonging to a Cff-structure. An almost contact structure (φ, ξ, θ) defines a hyperdistribution 3) and a complex structure / on 3). We shall also show that there is an affine connection with respect to which all structure tensors are parallel and whose torsion tensor is proportional to the Nijenhuis tensor formally defined by / when they are restricted to 3).In § 1, we shall recall definitions of a Ci?-structure and its integrability. Some facts about almost contact structures belonging to a Ci?-structure will also given. §2 will be devoted to the study of affine connections associated with almost contact structures. In §3, we shall obtain a change of canonical connections.The authors wish to express their hearty thanks to Professor S. Ishihara for his constant encouragement and valuable suggestions. § 1. CR-structures and almost contact structures.Let M be a connected C°°-manifold of dimension 2n + l (n^l). Let 3) denote a 1-codimensional subbundle of the tangent bundle TM, what is called a hyper-
IntroductionLet f:M~JV1 be an isometric immersion of a connected complete Riemannian manifold M into a Riemannian manifold/~. If for each geodesic ~ of M the curve f ~ in M has constant curvatures of osculating order d and they are independent of 7, then f is called a helical immersion of order d. In this paper we shall study a helical immersion into a unit sphere.In [6] Hong has studied a submanifold with planar geodesics in an Euclidean space and Little [8] and Sakamoto [13] classified planar geodesic immersions into a space form. Except for a totally geodesic case, such immersions are helical of order 2 (totally geodesic immersion is helical of order 1). Since for helical immersions of order 2 the second fundamental form is parallel, at present the classification of [13] is contained in that of Ferus [4]. Moreover Nakagawa [10] has studied a cubic geodesic immersion into a space form and when it is minimal, proper and the ambient manifold is a sphere, he showed that it is congruent to the standard immersion of a sphere which is constructed by spherical harmonic polynomials of degree 3. A Helical immersion of order 3 into a unit sphere is a proper cubic geodesic immersion. Therefore we conclude the following result: If f:M~S(1) is a helical minimal immersion of order 2 or 3, then M is a compact symmetric space of rank one and f is congruent to a standard minimal immersion given in [15].The concept "helical immersion" originates from Besse [3]. In this book, Besse showed that a strongly harmonic manifold admits a minimal isometric immersion into a sphere satisfying a nice property, which is the concept "helical". Harmonic manifolds have been studied by many authors (for example [1,12]). However the conjecture that harmonic manifolds are locally symmetric is still open. The theory of helical immersions (in particular, minimal case) is a submanifold version of harmonic manifolds. The author thus hopes that the study of helical immersions into a sphere will be useful to solve the above conjecture.In Sect. 1 we give basic notations and equations. Also Frenet formula of a curve in a Riemannian manifold will be explained. In Sect. 2, the definitions of 0025-5831/82/0261/0063/$03.60
In [2], S. Bochner defined a certain curvature tensor as a complex analogy of Weyl conformal curvature tensor without geometrical interpretation. At present this tensor is called Bochner curvature tensor. Recently Webster [9], [11] gave this geometrical interpretation as a pseudoconformal invariant on a Ci?-manifold. Indeed Bochner curvature tensor is the 4th curvature invariant given in Chern-Moser's paper [4] (cf. Tanaka [7]). In this paper we shall also derive Bochner curvature tensor from our argument of Ci?-structure.In [6] we studied almost contact structures standing on the viewpoint of pseudoconformal geometry and gave the change of canonical connections associated with almost contact structures belonging to the same Ci?-structure. The point under our discussion is the fact that almost contact structures belonging to a Ci?-structure play the same role as Riemannian structures belonging to a conformal structure and canonical connections correspond to Riemannian connections. Like the conformal change of Riemannian connections, a gradient vector appears in the change of canonical connections. Therefore we compute the difference of their curvature tensors and eliminate the gradient vector. Then we get a curvature invariant.In § 1 we recall definitions and results given in [6]. § 2 is devoted to the study of curvatures of canonical connections. We in § 3 obtain the curvature invariant of the pseudo-conformal geometry.The authors wish to express their hearty thanks to Professor S. Ishihara for his constant encouragement and valuable suggestions. § 1. Preliminaries.Let J be a connected orientable C°°-manifold of dimension 2n+l (n^l) and (£), J) a pair of a hyperdistribution 3) and a complex structure / on £). The pair (3), J) is called a CR-structure if the following two conditions hold: (c.i) zjχ,m-zχ, (C.2) ZJX, JYl-lX, Yl-ΛZX, JYl+ZJX, O=
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