Abstract. In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures. IntroductionAnalyzing the study of Riemannian geometry we see that its basic concepts are related with some operators, such as shape, Ricci, Schouten operator, etc. and functions constructed of them, such as mean curvature, scalar curvature, Gauss-Kronecker curvature, etc. The most natural and useful functions are the ones derived from algebraic invariants of these operators e.g. by taking trace, determinant and in general the r-th symmetric functions σ r . However, the case r > 1 is strongly nonlinear and therefore more complicated. The powerful tool to deal with this problem is the Newton transformation T r of an endomorphism A (strictly related with the Newton's identities) which, in a sense, enables a linearization of σ r , (r + 1)σ r+1 = tr (AT r ).Although this operator appeared in geometry many years ago (see, e.g., [21,29]), there is a continues increase of applications of this operator in different areas of geometry in the last years (see, among others, [1,2,3,8,10,17,18,23,24,25,28]).All these results cause a natural question, what happen if we have a family of operators i.e. how to define the Newton transformation for a family of endomorphisms. A partial answer to this question can be found in the literature (operator T r and the scalar S r for even r [5,15]), nevertheless, we expect that this case is much more subtle. This is because in the case of family of operators we should obtain more natural functions as in the case of one and consequently more information about geometry. In order to do this, for any multi-index u 2000 Mathematics Subject Classification. 53C12; 53C65.

Abstract. We study the geometry of a G-structure P inside the oriented orthonormal frame bundle SO(M ) over an oriented Riemannian manifold M . We assume that G is connected and closed, so the quotient SO(n)/G, where n = dim M , is a normal homogeneous space and we equip SO(M ) with the natural Riemannian structure induced from the structure on M and the Killing form of SO(n). We show, in particular, that minimality of P is equivalent to harmonicity of an induced section of the homogeneous bundle SO(M ) × SO(n) SO(n)/G, with a Riemannian metric on M obtained as the pull-back with respect to this section of the Riemannian metric on the considered associated bundle, and to the minimality of the image of this section. We apply obtained results to the case of almost product structures, i.e., structures induced by plane fields.

Let M be a submanifold of a Riemannian manifold (N,g). M induces a subbundle O(M,N) of adapted frames over M of the bundle of orthonormal frames O(N). The Riemannian metric g induces a natural metric on O(N). We study the geometry of a submanifold O(M,N) in O(N). We characterize the horizontal distribution of O(M,N) and state its correspondence with the horizontal lift in O(N) induced by the Levi–Civita connection on N. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold M with a deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures and compare this geometry with the geometry of M.

In this paper we study a geometric configuration of submanifolds of arbitrary codimension in an ambient Riemannian space. We obtain relations between the geometry of a q-codimension submanifold M n along its boundary and the geometry of the boundary Σ n−1 of M n as an hypersuface of a q-codimensional submanifold P n in an ambient space M n+q . As a consequence of these geometric ralations we get that the ellipticity of the generalized Newton transformations implies the tranversality of M n and P n in P n is totally geodesic in M n+q .

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