The partial Ricci curvature of a manifold inside a cylinder or cone in a Euclidean space
Let \(M^n\) be an \(n\)-dimensional manifold and let \(K\) be its Ricci curvature. By definition, put \(\text{Ric}^q (x_0; x_1,\dots ,x_q)=\sum_{i=1}^{q}K(x_0,x_i)\) for arbitrary \(q+1\) mutually orthogonal norm-one vectors \(x_0, x_1, \dots ,x_q\) in \(TM^n\). The function \(\text{Ric}^q\) is a kind of an interpolation between the sectional and Ricci curvatures and is referred to as the Ricci \(q\)-curvature or as the partial Ricci curvature. This function was introduced by \textit{H. Wu} [Indiana Univ. Math. J. 36, 525--548 (1987; Zbl 0639.53050)] and studied by \textit{Z. Shen} [Trans. Am. Math. Soc. 338, 289--310 (1993; Zbl 0783.53026)]. In the article under review, the author studies interrelations between the partial Ricci curvature and the exterior geometry of a low-codimensional submanifold which lies inside a cylinder or cone in a Euclidean space. He derives lower bounds for the radius of an enveloping cylinder and for the altitude of the manifold inside a cone. The estimates obtained depend on the codimension of the manifold and on an upper bound on the partial Ricci curvature of the manifold. A typical result is as follows. Let \(M^n\) be a compact \(n\)-dimensional manifold and let \(\text{Ric}^q\) be greater or equal to \(qc^2\) for some \(q\in [1,n-1]\) and \(c=\text{const}>0\). Then \(M^n\) cannot be isometrically embedded into \(\mathbb R^{n+p}\) inside a circular cylinder with \((2p+q-1)\)-dimensional parallel and radius \(r=1/c\). The established results generalize theorems previously obtained by \textit{S.-S. Chern} and \textit{N. H. Kuiper} [Ann. Math. (2) 56, 422--430 (1952; Zbl 0052.17601)], by \textit{T. Otsuki} [Proc. Japan Acad. 29, 99--100 (1953; Zbl 0052.17602)], and by \textit{H. Jacobowitz} [Proc. Am. Math. Soc. 40, 245--246 (1973; Zbl 0265.53047)].
scientific article; zbMATH DE number 1285374