A method is proposed for constructing minimal surfaces based on fifth-order Bezier isotropic curves specified in a vector-parametric form, allowing control of the guide curve and the surface in user mode. The coefficients of the basic quadratic forms were calculated and it was shown that the surfaces would be minimal. An example of a surface constructed by the proposed method is given.

Under certain assumptions, we characterize the almost $\eta$-Einstein, $\eta$-Ricci and $\eta$-Yamabe solitons on a pseudo-Riemannian manifold when the potential vector field of the soliton is infinitezimal harmonic or torse-forming. Moreover, in the second case, if the manifold is Ricci symmetric of constant scalar curvature, then the soliton is completely determined.

By applying an average method in PDE, we obtain a dichotomy between “constancy” and “infinity” of the warping functions on complete noncompact Riemannian manifolds for an appropriate isometric immersion of a multiply warped product manifold $N_1\times_{f_2} N_2 \times \cdots \times _{f_k} N_k\, $ into a Riemannian manifold. Generalizing the earlier work of the authors in [9], we establish sharp inequalities between the mean curvature of the immersion and the sectional curvatures of the ambient manifold under the influence of quantities of a purely analytic nature (the growth of the warping functions). Several applications of our growth estimates are also presented.

In the case when $M$ is equipped with a bi-Hamiltonian structure $(M,\pi_1, \pi_2)$ we show how to construct family of Poisson structures on the tangent bundle $TM$ to a Poisson manifold. Moreover we present how to find Casimir functions for those structures and we discuss some particular examples.

The problem of a fluid body rotating with a constant angular velocity and subjected to uniform external pressure is of real interest in both fluid dynamics and nuclear theory. Besides, from the geometrical viewpoint the sought equilibrium configuration of such system turns out to be equivalent to the problem of determining the surface of revolution with a prescribed mean curvature. In the simply connected case, the equilibrium surface can be parameterized explicitly via elliptic integrals of the first and second kind.