  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 $C^1$-estimate for solutions to the
Dirichlet problem for $m$-Hessian equations.
