On totally geodesic foliations with bundle-like metric

Abstract

Let F be a totally geodesic foliation of dimension n and codimension p on a Riemannian manifold (M, g). Suppose that g is a bundle-like metric for F and M has at least one point at which none of its mixed sectional curvatures vanishes. Under these conditions we prove that n ≤ p − 1. We show that this inequality is optimal, and none of the above conditions can be removed. (2000): 53C12, 53C20. Mathematics Subject Classification

Curve flows in Lagrange–Finsler geometry, bi-Hamiltonian structures and solitons

“…This approach can be developed into a geometric formalism for mapping arbitrary (semi) Riemannian metrics [22] and regular Lagrange mechanical systems into bi-Hamiltonian structures and related solitonic equations following certain methods elaborated in the geometry of generalized Finsler and Lagrange spaces [2,3,23] and nonholonomic manifolds with applications in modern gravity [24,25,26].…”

“…Remark 2.1 A manifold (or a bundle space) is called nonholonomic if it is provided with a nonholonomic distribution (see historical details and summary of results in [24]). In a particular case, when the nonholonomic distribution is of type (3), such spaces are called N-anholonomic [26].…”

Nonholonomic Ricci Flows and Running Cosmological Constant I: 4d Taub-Nut Metrics

“…Such spacetimes are distinguished by corresponding nontrivial nonholonomic (foliated, if the integrability conditions are satisfied) structures (examined in different approaches to Lagrange-Finsler geometry [33], semi-Riemannian and Finsler foliations [36] and, for instance, in modern gravity and noncommutative geometry [15]). In searching for physical applications of such geometric methods, we addressed to the geometry and physics of Taub-NUT spacetimes [1,2,3,4,5,6,7] (see also more recent developments in Refs.…”

On Paraquaternionic Submersions Between Paraquaternionic Kähler Manifolds

“…We recall the following well-known result (see, for example, [3]). , and two open submanifolds U and U ⊥ respectively of the leaf of F and of F ⊥ passing through x * , such that, denoting with g and g ⊥ the metrics naturally induced on U and U ⊥ , (U,g| U ) is the semi-Riemannian product of (U, g) and (U ⊥ , g ⊥ ).…”

“…It is well known ([1], [3], [6]) that the assignment of two complementary distributions D and D ′ on a manifold M is equivalent to the assignment of an almost product structure on it, that is a (1, 1)-type tensor field F = ±I, satisfying F 2 = I, with D = T + M and D ′ = T − M , where T + M and T − M are the eigendistributions with respect the eigenvalues ±1 of F . The integrability of both distributions is equivalent to the vanishing of the Nijenhuis tensor field N F related to the structure F (see [20]), and in this case the tensor field F is called a product structure, or a locally product structure.…”