A New Fractional Curvature of Curves and Surfaces in Euclidean Space Using the Caputo’s Fractional Derivative

In this paper, the authors generalize the fractional curvature of plane curves introduced by Rubio et al. in 2023, to regular curves in the Euclidean space (Formula presented.), and study the geometric properties of the curve using Caputo’s fractional derivative. Furthermore, we introduce a new definition of fractional curvature and fractional mean curvature of a regular surface, using fractional principal curvatures; and prove that such concepts are invariant under isometries; i.e., they belong to the intrinsic geometry of the regular surface. Also, a geometric interpretation is given to Caputo’s fractional derivative of algebraic polynomials.