After Thomas James Willmore, many authors were looking for an immersion of a manifold in Euclidean space or Riemannian manifold, which is the critical point of functionals whose integrands depend on the mean curvature or the norm of the second fundamental form. We study a new Willmore-type variational problem for a foliated hypersurface in Euclidean space. Its general version is the Reilly-type functional, where the integrand depends on elementary symmetric functions of the eigenvalues of the restriction on the leaves of the second fundamental form. We find the 1st and 2nd variations of such functionals and show the conformal invariance of some of them. For a critical hypersurface with a transversally harmonic foliation, we derive the Euler-Lagrange equation and give examples with low-dimensional foliations. We present critical hypersurfaces of revolution and show that they are local minima for special variations of immersion.

Einstein considered a linear connection $\mathrm{\nabla }$ with torsion $T$ on a differentiable manifold equipped with a nonsymmetric (0, 2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric associated with gravity, and $F\ne 0$ is a skew-symmetric tensor associated with electromagnetism, such that $({\mathrm{\nabla }}_{X}G)(Y,Z)=-G(T(X,Y),Z)$. In this paper, we explicitly present the Einstein connection of a nonsymmetric pseudo-Riemannian manifold with non-degenerate $F$, satisfying the $f^2$-torsion condition $T(f^{2}X,Y)=T(X,f^{2}Y)=f^{2}T(X,Y)$, where $g(X,fY)=F(X,Y)$, and show that in the almost Hermitian case, it reduces to the Prvanović's (1995) solution. We also explicitly present the Einstein connection of almost contact metric manifolds satisfying the $f^2$-torsion condition, discuss special Einstein connections, and give example in terms of the weighted product of almost Hermitian manifolds.

We study so-called "weak" metric structures on a smooth manifold, which generalize the metric contact and $K$-contact structures and allow a new look at the classical theory. We characterize weak $K$-contact manifolds among all weak contact metric manifolds using the property well known for $K$-contact manifolds, as well as find when a Riemannian manifold endowed with a unit Killing vector field is a weak $K$-contact manifold. We also find sufficient conditions for a weak $K$-contact manifold with a parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.