A best possible inequality for curvature-like tensor fields.

*(English)*Zbl 1175.53023In the differential geometry of submanifolds relations between intrinsic and extrinsic curvature invariants are of particular interest. In [Arch. Math. 60, No. 6, 568–578 (1993; Zbl 0811.53060)], B.-Y. Chen introduced a new intrinsic curvature invariant, namely
\[
\delta(p) : = (\tau - \inf \kappa)(p)
\]
at a point \(p\); here \(\tau\) is the scalar curvature and \(\kappa\) the sectional curvature. He studied relations of this invariant with other curvature invariants. His idea was modified and applied to different geometric situations by several authors.

The authors of the present paper study algebraic curvature tensor fields on a Riemannian manifold \((M,g)\) and extend the definition of Chen to such tensor fields, moreover they consider special algebraic curvature tensors induced by symmetric \((1,2)\)-tensor fields, imitating the typical algebraic curvature type terms defined via the cubic form, appearing in the integrability conditions (in terms of the relative metric) of relative hypersurface theory and also in the theory of Lagrangian submanifolds. The main result of this paper is an inequality between the Chen type invariant and a Pick type invariant within the concept of the authors. They discuss the case of equality. Finally they apply their results to Lagrangian submanifolds and in centroaffine hypersurface theory.

Reviewer’s remarks. 1. In the literature, the standard terminology for the author’s terminology “curvature-like tensor fields” is “algebraic curvature tensors”, see e.g. pp. 45–48 in [A. L. Besse, Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 10. Berlin etc.: Springer-Verlag (1987; Zbl 0613.53001)]. 2. Theorem 6 in the author’s paper has a nice geometric consequence: The hypersurface is an affine hypersphere from the coincidence of the centroaffine with the Blaschke normal.

The authors of the present paper study algebraic curvature tensor fields on a Riemannian manifold \((M,g)\) and extend the definition of Chen to such tensor fields, moreover they consider special algebraic curvature tensors induced by symmetric \((1,2)\)-tensor fields, imitating the typical algebraic curvature type terms defined via the cubic form, appearing in the integrability conditions (in terms of the relative metric) of relative hypersurface theory and also in the theory of Lagrangian submanifolds. The main result of this paper is an inequality between the Chen type invariant and a Pick type invariant within the concept of the authors. They discuss the case of equality. Finally they apply their results to Lagrangian submanifolds and in centroaffine hypersurface theory.

Reviewer’s remarks. 1. In the literature, the standard terminology for the author’s terminology “curvature-like tensor fields” is “algebraic curvature tensors”, see e.g. pp. 45–48 in [A. L. Besse, Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 10. Berlin etc.: Springer-Verlag (1987; Zbl 0613.53001)]. 2. Theorem 6 in the author’s paper has a nice geometric consequence: The hypersurface is an affine hypersphere from the coincidence of the centroaffine with the Blaschke normal.

Reviewer: Udo Simon (Berlin)