Showing papers in "Kodai Mathematical Seminar Reports in 1971"

On the concurrent vector fields of immersed manifolds

Abstract: Let Rm be an m-dimensional Riemannian manifold1:> with covariant derivative D and let x: M n-*Rm be an immersion of an ^-dimensional manifold Mn into Rm. A vector field X in Rm over Mn is called a concurrent vector field^ if we have dx+DX=Q, where dx denotes the differential of the immersion x. In particular, if X is a normal vector field of Mn in Rm, then the vector field X is called a concurrent normal vector field. The main purpose of this paper is to study the behavior of the concurrent vector fields of immersed manifolds and also find a characterizati on of the concurrent vector fields with constant length.

On normal (f, g, u, v, λ)-structures on submanifolds of codimension 2 in an even-dimensional Euclidean space

Abstract: It is well known that a hypersurface of an almost Hermitian manifold admits an almost contact metric structure naturally induced on it. The study of hypersurfaces of a Euclidean space and of a Kahlerian manifold on which the induced almost contact metric structure satisfies certain conditions has been started by one of the present authors [4, 5]. On the other hand Blair [1, 2], Goldberg [3], Ludden [1, 2], Yamaguchi [8] and the present authors [6, 9] started the study of hypersurface of an almost contact manifold and of submanifolds of codimension 2 of an almost complex manifold. These submanifolds admit, under certain conditions, what we call (/, g, u, v, λ) structure. An even-dimensional sphere of codimension 2 of an even-dimensional Euclidean space is a typical example, of a manifold which admits this kind of structure. In a previous paper [9], we have studied the (/, g, u, v, -structure and given characterizations of even-dimensional sphere. In the present paper, we study submanifolds of codimension 2 in an evendimensional Euclidean space which admit a normal (/, g, u, v, Λ)-structure. In § 1, we consider submanifolds of codimension 2 of an even-dimensional Euclidean space regarded as a flat Kahlerian manifold. In the next section, we deal with (/, g, u, v, -structure induced on a submanifold of codimension 2 of an even-dimensional Euclidean space. In § 3, we find differential equations which /, g, u, v and λ satisfy. § 4 is devoted to the study of relations between the structure equations of the submanifold and the induced (/, g, u, v, ^-structure. In § 5 we prove a series of lemmas which are valid for normal (/, g, u, v, λ}structures and in § 6 we study properties of the mean curvature vector of the submanifold with normal (/, g, u, v, -structure. In the last § 7, we study hypersurfaces of an odd-dimensional Euclidean space and determine all the hypersurfaces admitting a normal (/, g, u, v, Λ)-structure. Our main theorem appears at the end of §7.

Univalency of analytic mappings of a Riemann surface into itself

Abstract: 1. In the present paper we shall study a Riemann surface whose every nonconstant analytic mapping into itself is univalent. Let S be the class of Riemann surfaces whose every non-constant analytic mapping into itself is univalent, and let K be the class of Riemann surfaces whose every non-constant analytic mapping into itself is univalent and onto. It is easy to see that φ^K^SdOABΓiff where H is the class of Riemann surfaces whose universal covering are conformally equivalent to the unit disk. Heins [5] showed OGΓ[Hc:S and KGdK where KG denotes the class of Riemann surfaces with a finite positive genus or with a finite number of planar boundary elements belonging to OG Π H. Kubota [8] introduced a class of Riemann surfaces and showed that the class is a subclass of K. In § 2 we construct an example of Riemann surface of class OAB^H on which there exists a non-univalent analytic mapping into itself. Namely we show S^OABΓ\PL In §3 we introduce a class KΠD of Riemann surfaces and show KHB^KHDCLK, where KHB denotes the class of Riemann surfaces introduced by Kubota. Heins [5] showed that if W is of class KG and of finite genus, then the number of non-constant analytic mappings of W into itself is finite. In § 4 we show the same result with respect to a Riemann surface of class KHD

Distribution and critical curves in Riemannian manifold

Abstract: Let S) be a C°° distribution in a C°° Riemannian manifold M. In the present paper a curve of M where every tangent vector lies in $) is called a 3) -curve. Let P and Q be two points of M such that there exist ^)-curves joining P and Q. We call a S) -curve C a critical 2) -curve with the fixed end points P, Q if the length I of C takes a critical value in the set of £p-curves joining P and Q. The purpose of the present paper is to find differential equations of critical £7)-curves when n—m=dim g) satisfies n<2(n—m\\ where n=dim M, and to study properties of such critical $) -curves in some special cases.

Pseudo-umbilical surfaces in euclidean spaces

Abstract: BY BANG-YEN CHENRecently, the author introduced the notion of αth curvatures of first and secondkinds for surfaces in higher dimensional euclidean space [2, 3]. The main purposeof this paper is to study these curvatures more detail. In §1, we derive someintegral formulas for the αth curvatures of first and second kinds. In §2, we getsome applications of these formulas to pseudo-umbilical surfaces.