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Having in mind the well known model of Euclidean convex hypersurfaces [4], [5], and the ideas in [1] many authors defined and investigate convex hypersurfaces of a Riemannian manifold. As it was proved by the first author in [7], there follows the interdependence between convexity and Gauss curvature of the hypersurface. In this paper we define H-convexity of a Riemannian submanifold of arbitrary codimension, replacing the normal versor of a hypersurface with the mean curvature vector. A characterization, used by B.Y. Chen [2], [3] as the definition of strictly H-convexity, it is obtained.
Based on stochastic curvilinear integrals in the Cairoli-Walsh sense and in the Itô-Udrişte sense, we develop an original theory regarding the multitime stochastic differential systems. The first group of the original results refer to the complete integrable stochastic differential systems, the path independent stochastic curvilinear integral, the Itô-Udrişte stochastic calculus rules, examples of path independent processes, and volumetric processes. The second group of original results include the multitime Itô-Udrişte product formula, first stochastic integrals and adjoint multitime stochastic Pfaff systems. Thirdly, we formulate and we prove a multitime maximum principle for optimal control problems based on stochastic curvilinear integral actions subject to multitime Itô-Udrişte process constraints. Our theory requires the Lagrangian and the Hamiltonian as stochastic 1-forms.
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