This paper is devoted to the linear analysis of a slender homogeneous piezoelectric beam that undergoes tip loading. The solution of the Saint-Venant’s problem presented in this paper generalizes the known solution for a homogeneous elastic beam. The analytical approach in this study is based on the Saint-Venant’s semi-inverse method generalized to electroelasticity, where the stress, strain, and (electrical) displacement components are presented as a set of initially assumed expressions that contain tip parameters, six unknown coefficients, and three pairs of auxiliary (torsion/bending) functions in two variables. These pairs of functions satisfy the so-called coupled Neumann problem (CNP) in the cross-sectional domain. In the limit “elastic” case the CNP transforms to the Neumann problem, for a beam made of a poled piezoceramics the CNP is decomposed into two Neumann problems. The paper develops concepts of the torsion/bending functions, the torsional rigidity and shear center, the tip coupling matrix for a piezoelectric beam. Examples of exact and numerical solutions for elliptical and rectangular beams are presented.

A pseudo-Riemannian manifold endowed with k > 2 orthogonal complementary distributions (called a Riemannian almost multi-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, Dupin hypersurfaces and in studies of the curvature and Einstein equations. In this article, we consider the following two problems on the mixed scalar curvature of a Riemannian almost multi-product manifold with a linear connection: a) integral formulas and applications to splitting of manifolds, b) variation formulas and applications to the mixed Einstein-Hilbert action, and we generalize certain results on the mixed scalar curvature of pseudo-Riemannian almost product manifolds.

We examine the total mixed scalar curvature of a smooth manifold endowed with a distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution and use this key and technical result to find the critical points of this action. Together with the arbitrary variations of the metric, we consider also variations that preserve the volume of the manifold or partially preserve the metric (e.g., on the distribution). For each of those cases, we obtain the Euler-Lagrange equation and its several solutions. Examples of critical metrics that we find are related to various fields of geometry such as contact and 3-Sasakian manifolds, geodesic Riemannian flows, codimension-one foliations, and distributions of interesting geometric properties (e.g., totally umbilical and minimal).

We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained Euler-Lagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.

For every globally hyperbolic spacetime M, we derive new mixed gravitational field equations embodying the smooth Geroch infinitesimal splitting T (M) = D ⊕ R∇T of M, as exhibited by Bernal and Sánchez (2005 Commun. Math. Phys. 257 43-50). We give sufficient geometric conditions (e.g. T is isoparametric and D is totally umbilical) for the existence of exact solutions −β dT ⊗ dT + g to mixed field equations in free space. We linearize and solve the mixed field equations Ric D (g) μν − ρ D (g) g μν = 0 for empty space, where ρ D (g) is the mixed scalar curvature of foliated spacetime (M, D) (due to Rovenski (2010 arXiv:1010.2986 v1[math.DG])). If g = g 0 + γ is a solution to the linearized field equations, then each leaf of D is totally geodesic in (R 4 \ R, g ) to order O( ). We derive the equations of motion of a material particle in the gravitational field g μν governed by the mixed field equations Ric D (g) μν − ρ D (g) ω μ ω ν − g μν = 2πκc −2 T μν − 1 2 T g μν . In the weak field ( 1) and low velocity ( v /c 1) limit, the motion equations are d 2 r/dt 2 = ∇φ + F, where φ = ( /2)c 2 γ 00 .

We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their L 2 adjoint operators on tensors. Then, we introduce the Weitzenböck type curvature operator on tensors, prove the Weitzenböck type decomposition formula, and derive the Bochner–Weitzenböck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones.

In this article, we generalize known integral formulae (due to BritoLangevin-Rosenberg, Ranjan and the second author) for foliations of codimension 1 or unit vector fields and obtain an infinite series of such formulae involving invariants of the Weingarten operator of a unit vector field, of the Jacobi operator in its direction, and their products. We write several such formulae explicitly, on locally symmetric spaces as well as on arbitrary Riemannian manifolds where they involve also covariant derivatives of the Jacobi operator. We work also with foliations of codimension 1 (or vector fields) which admit "good" (in a sense) singularities.

We study the geometry of the weak almost para-f -structure and its satellites. This allow us to produce totally geodesic foliations and Killing vector fields and also to take a fresh look at the para-f -structure introduced by A. Bucki and A. Miernowski. We demonstrate this by generalizing several known results on almost para-f -manifolds. First, we express the covariant derivative of f using a new tensor on a metric weak para-f -structure, then we prove that on a weak para-K-manifold the characteristic vector fields are Killing and ker f defines a totally geodesic foliation. Next, we show that a para-S-structure is rigid (i.e., a weak para-S-structure is a para-S-structure), and that a metric weak para-f -structure with parallel tensor f reduces to a weak para-C-structure. We obtain corollaries for p = 1, i.e., for a weak almost paracontact structure.