A general description of totally geodesic foliations

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We give some general concepts and results for totally geodesic foliations &~ on complete Riemannian manifolds. In particular, we reduce the problem to that of a generalization of the theory of principal connections. This enables us to show that the global geometry of ^~ is related to certain sheaves of germs of local Killing vector fields for the Riemannian structure along the leaves. Further, we define a cohomology group H?g, and natural mappings from Ht* into the de Rham cohomologies of the leaves, such that the characteristic classes in Ht*g are mapped to the characteristic classes of the leaves. Introduction. A foliation _ 7 on a Riemannian manifold M, is said to be totally geodesic if each geodesic of M is everywhere or nowhere tangent to ^ 7 The different aspects of such foliations have been examined by many authors; for different approaches and more references, see for instance [14], [10], [1] and [4]. In particular, the codimension one case has been classified [12], and there is a homological classification of the dimension one case [28]. The purpose of this paper is to apply, to totally geodesic foliations of arbitrary dimension, the techniques developed in the theory of Riemannian foliations, and in particular, the works of Molino [16] [21]. Recall that a foliation ^ on a manifold M is Riemannian if there exists a Riemannian metric on M such that ^ can be defined by local Riemannian surmersions [23]. Equivalently, ^~ is Riemannian if M possesses a Riemannian metric whose geodesies are everywhere or nowhere perpendicular to ^ Ί Riemannian foliations have been extensively studied, and in particular, there is a strong structure theorem [20]. Given the evident analogy between totally geodesic and Riemannian foliations, it is not surprising that there are many concepts and results from the Riemannian case that find similar expression in the totally geodesic situation. Indeed, by pursuing this approach one obtains a good geometric description and a useful cohomology group. A first application of this work to the dimension ^ 3 cases is given in [7]. In collaboration with E. Ghys, we have given a detailed account of totally geodesic foliations on 4-manifolds [9].