A new intrinsic curvature invariant for centroaffine hypersurfaces
The authors introduce the invariant \(\varepsilon _M\) for definite centroaffine hypersurfaces \(M^n \rightarrow {\mathbb R}^{n+1}\) in terms of the normalized scalar curvature of a plane section \(\pi \) in \(T_pM\). They derive the inequality \[ \varepsilon _M \geq -{\textstyle{\frac{n^2(n-2)}{2(n-1)}}}h(T^{\# },T^{\# }) + {\textstyle{\frac{1}{2}}}(n+1)(n-2), \tag{*} \] where \(h\) is a positive definite metric and \(T^{\# }\) the Tschebychev field. The class of hypersurfaces satisfying the equality in (*) is a very wide one. They prove that this class consists a proper affine hyperspheres centered at the origin. Using some distribution \(D\) on such hypersurfaces, the authors obtain classification theorems under the additional assumption that either \(M^n\) has constant scalar curvature or the distribution \(D^{\perp }\) is integrable. Examples related to equiaffine elliptic spheres in \({\mathbb R}^3\), realizing the equality in (*) are also given. Some results about polarization and asymptotic spectral invariants of second order Laplace type operators are related to the previously mentioned results.
asymptotic spectral geometry of Laplace type operators
scientific article; zbMATH DE number 1066112