121625805
Recurrent tensors on a linearly connected differentiable manifold
Introduction. Let M be a connected smooth (i.e., CI) manifold of dimension n, and B the total space of the frame bundle over M. Let a linear connection be given on M, and let B [zO] be the submanifold of B consisting of all the points which can be joined to a given point zo in B by sectionally smooth horizontal curves. The purpose of this paper is to elaborate on a natural correspondence set up by S. S. Chern [2, pp. 78-79] between tensors of type (r, s) on M and sets of nr+, functions of a particular type on B, and use it to prove the theorem that a tensor S on M is recurrent (i.e., S is not a zero tensor and its covariant derivative is equal to the tensor product of a covariant vector and S itself) if the restrictions to B [zo] of its corresponding functions on B have no common zero and are proportional to a set of constants. We give several applications of this theorem, obtaining among others the following results: (i) A recurrent tensor on M has no zero (Theorem 3.8); (ii) A property obtained by K. Nomizu [4, p. 73] characterizing linear connections with covariantly constant curvature tensor or covariantly constant torsion tensor (Theorem 4.3); (iii) A property characterizing linear connections with recurrent curvature or recurrent torsion similar to (ii) above (Theorem 4.2); (iv) The holonomy group of a linear connection with recurrent curvature is at most of dimension n(n1)/2 (Corollary 4.4). The author is deeply indebted to Professor S. S. Chern for his generous help during the first half of the year 1959 at the University of Chicago when the author tried to acquaint himself with the modern theory of linear connections. The author is also grateful to the referee for pointing out a gap in the original proof of our main theorem (3.9) and for his valuable suggestions which have resulted in several improvements in this paper. 1.1. Bundle of frames. In ? ?1.1-1.5, we summarize some of the results on linear connections which are needed for our later work. We assume as known the classical theory of linear connections and the elementary properties of n-dimensional smooth manifolds (class Cand with countable base). All the vector fields, tensor fields, q-forms, etc. defined on a smooth manifold or on part of it are assumed to be of class Co unless stated otherwise. Each of the
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A note on Levine’s paper: “Fields of parallel vectors in projectively flat spaces”
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Linear connections and quasi-connections on a differentiable manifold
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LINEAR CONNEXIONS WITH ZERO TORSION AND RECURRENT CURVATURE
Introduction. Let M be a connected n-dimensional C4-manifold. If S is any tensor (field) on M, then (i) S(u) denotes the value of S at the point uGM; (ii) S =0 means that S is everywhere zero, i.e. S(u) =0 for every u CM; (iii) S $0 means that S is not everywhere zero, i.e. S(u) 0 for some u CM; (iv) S$O means that S is nowhere zero, i.e. S(u) =0 for no uCM. Throughout this paper, a tensor of type (1, 0) will be called a vector, and a tensor of type (0, 1) a covector. Summation over a repeated index, Latin or Greek, is always implied. Let y be a linear connexion on M. If rFJh (1_ a, h, i, j, * , ?n) are the components of y in the local coordinate system (U, uh) in M, then the components in (U, uh) of the torsion tensor T and the curvature tensor R on M are respectively
6405336
SECTIONAL CURVATURES OF GRASSMANN MANIFOLDS
(1) Introduction.-Let F be the field R of real numbers, the field C of complex numbers, or the field H of real quaternions; Fn+m a left (n + m)-dimensional Hermitian vector space over F; and Gn(FP+m) the Grassmann manifold of nplanes in F"+m provided with the invariant Riemannian metric with respect to which the distance between two points A and B in G.(Fn +m) is equal to the square root of the sum of the squares of the angles between the n-planes A and B in Ff +n (see ref. 14). Previous studies of G,(Fn +m) have not unearthed sufficiently precise information about its sectional curvatures. Although the components of the curvature tensor at a point of GC(Rn+") and Gn(Cn+") have been computed in references 5, 8, and 10, all that is so far known about the sectional curvatures of G.(F +1") seems to be that they are nonnegative but not all positive unless min(n,m) = 1, in which case they have the range of values given in Theorem 1 in §(4) (see ref. 2, p. 171; ref. 3, Theorem 7; ref. 1, p. 59; ref. 9, pp. 351 and 358; and ref. 8, Theorem 4.5). In this note, which is a continuation of reference 14 and a companion to reference 15, we shall give a complete description of the sectional curvatures of Gn(F"+m) In § § (2) and (3), we express the curvature tensor and the sectional curvature of G, (Fn+m) in terms of local coordinates and define the unitary curvature of Gn(Hn+m). In §(4), we state our main results concerning the range of values of the sectional curvature and certain characteristic properties of sections of minimum and maximum curvatures. Details of these results and similar results for the classical bounded symmetric domains will be published later. (2) The Curvature Tensor.-We know (ref. 14) that the invariant Riemannian metric on Gn(Fn+m) can be arrived at in a natural and geometric way; moreover, in a typical local coordinate system (U,Z) with neighborhood U and coordinate Z (which is an n X m matrix with elements in F), it has the explicit expression
176838628
Erratum to: “On an explicit characterization of spherical curves”
119676639
Reflexivity of Locally Convex Riesz Spaces