Significance of Hessian operators part5(Machine Learning)
- Differential geometry in the theory of Hessian operators(arXiv)
Author : N. M. Ivochkina, N. V. Filimonenkova
Abstract : The paper introduces a new differential-geometric system which originates from the theory of m-Hessian operators. The core of this system is a new notion of invariant differentiation on multidimensional surfaces. This novelty gives rise to the following absolute geometric invariants: invariant derivatives of the surface position vector, an invariant connection on a surface via subsurface, curvature matrices of a hypersurface and its normal sections, p-curvatures and m-convexity of a hypersurface, etc. Our system also produces a new interpretation of the classic geometric invariants and offers new tools to solve geometric problems. In order to expose an application of renovated geometry we deduce an a priori C1-estimate for solutions to the Dirichlet problem for m-Hessian equations
