Let Rm be an m-dimensional Riemannian manifold1:> with covariant derivative. D and let x: Mn-*Rm be an immersion of an ^-dimensional manifold Mn into Rm.
KODAI MATH. SEM. REP. 23 (1971), 343—350
ON THE CONCURRENT VECTOR FIELDS OF IMMERSED MANIFOLDS BY KENTARO YANO AND BANG-YEN CHEN Let Rm be an m-dimensional Riemannian manifold1:> with covariant derivative D and let x: Mn-*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 characterization of the concurrent vector fields with constant length. § 1. Preliminaries. Let Rm be an m-dimensional Riemannian manifold with covariant derivative D. By a frame eι, ~,em, we mean an ordered set of m orthonormal vectors eι, ,em in the tangent space at a point of Rm. The frame e^ •••, em defines uniquely a dual coframe ω1} •••, ωm in the cotangent space and vice versa. The fundamental theorem of local Riemannian geometry says that in a neighborhood U of a point p there exists uniquely a set of linear differential forms ωAB satisfying the conditions: (1)
and (2)
where here and in the sequel the indices A, B, ••• run over the range {1, •••, m}. The linear differential forms WAB are called the connection forms and the covariant derivative DX of a vector field -X"=Σ-X^> is given by Received October 31, 1970. 1) Manifolds, mappings, tensor fields and other geometric objects are assumed to be differentiate and of class C°°. 2) In [3], a vector field X is called a concurrent vector field if there exists a function / such that dx+D(fX)=Q, but in this paper, we adopt the above definition.
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(3)
where (4)
The vector field X is said to be parallel if DX=Q. themselves equation (3) gives (5)
For the vectors eA
DeA = ΣωAB®en. The structure equations of Rm are given by (2) and
( 6)
daJAB= ΣAC/\ωcB + ΩAB,
ΩAB = Σ -9-
The tensor field RABCD is called the Riemann-Christoffel tensor. From RABCD the Ricci tensor and the scalar curvature are defined respectively by (7)
RAB — RBA — Σ RCABC,
S
(8)
Let x: Mn-*Rm be an immersion of an ^-dimensional manifold Mn into Rmt and let B be the set of all elements b=(p, elt •••, en, en+ι, •••, 0m) such that^€M Λ , 0ι, — ,e n are tangent vectors, en+ι,—,em are normal vectors to x(Mn) at ar(ί) and (x(p\ ei, — , ^w) is a frame in J?m, where we have identified dx(βi) with ^ f=l, — , «. Letft>^,WAB be the forms previously denoted by ωA, ωAB relative to this particular frame field. Then we have (9)
ωr=0,
r, s, t, - • = »+!, ••-, m.
Taking the exterior derivative and using (2), we get (10)
Σω
