Horizontal Newton operators and high-order Minkowski formula

Abstract

In this paper, we study the horizontal Newton transformations, which are nonlinear operators related to the natural splitting of the second fundamental form for hypersurfaces in a complex space form. These operators allow to prove the classical Minkowski formulas in the case of real space forms: unlike the real case, the horizontal ones are not divergence-free. Here, we consider the highest order of nonlinearity and we will show how a Minkowski-type formula can be obtained in this case.

Minkowski identities for hypersurfaces in constant sectional curvature manifolds

“…In the first references above, the notion of the position vector field P is quite obscure and the attempted generalization of the Minkowski identities to constant sectional curvature ambient M yields a different result from what is found today in [5] and [6]. The latter recently discovered 'Hsiung-Minkowski' identities are proved again in the present article.…”

“…In Corollary 4.1 below we generalize the construction of local position vector fields. The next theorem is found in the works of P. Guan and J. Li [5] and C. Guidi and V. Martino [6]. In their proofs, the authors recur to Newton's identities for symmetric polynomials, which play a central role just like they played originally in Reilly's proof, in [11], of the Hsiung-Minkowski identities for Euclidean space.…”

“…A position vector field P exists on M and yields a horizontal λ c -mirror vector field π * P . We refer to [5,6,10] for the previous assertion, where we see that P in polar coordinates is given by P = s c (r)∂ r , with r the geodesic distance function from a base point in M . Such vector field P satisfies, ∀Y ∈ T M ,…”

New Hsiung-Minkowski identities

“…B. Chen and K. Yano in [4] also gave the most general integral formulas, for closed submanifolds of higher codimension in Euclidean space. Regarding the other space forms, P. Guan and J. Li [5] and C. Guidi and V. Martino [6] gave independent proofs of (2) recurring to Newton's identities for symmetric polynomials. In the same trend the new formulas of K. Kwong [9,10] have appeared.…”

Horizontal curvatures and classification results

“…These kind of horizontal curvatures appear quite naturally and have been recently studied to obtain integral formulas of Minkowski type [13,5]. Let us recall also the following definitions.…”