An application of P. A. Shirokov's theorem to the Bochner technique
Let \((M,g)\) be an \(n\)-dimensional compact Riemannian manifold, \(K\) its sectional curvature, and \(\omega_p\) a \(p\)-form on \(M\). The author obtains a symmetric 2-tensor field \(G\) from \(\omega_p\) and \(g\). The aim of this paper is to determine under which conditions \((M,g)\) (with \(K\) nonpositive) is reducible or irreducible and \((M,g)\) carries a non-zero Killing (or flat) form \(\omega_p\), \(1<p<n\). Let \((x^i)\) be local coordinates in \(M\), \(({ \partial \over \partial x^i})\) the natural frame, \(g_{ij}= g({\partial \over\partial x^i}, {\partial \over \partial x^j})\) and \(\omega_{i_1 \dots i_p}\) the components of \(\omega_p\). If \(\omega_p\) is a Killing \(p\)-form of components \(\omega_{i_1 \dots i_p}\), and \(G_{ij}= \omega_{ii_2 \dots i_p} \omega_j^{i_2 \dots i_p}\), then (*) \(G_{ij}= {1\over 2}\| \omega \|^2g_{ij}\). Theorem. Assume that an \(n\)-dimensional compact oriented Riemannian manifold \((M,g)\) carries a non-zero Killing form \(\omega_p\), \(1<p<n\). (1) If \(K<0\) everywhere on \((M,g)\), then \(\omega_p\) has maximal rank and satisfies (*). (2) If \(K\leq 0\) everywhere on \((M,g)\), then \(\| \omega \| =\text{const.}\) and either \((M,g)\) is reducible or \(\omega_p\) has maximal rank and satisfies (*). The method is based on the well-known theorem of \textit{P. A. Shirokov} [Izv. fiz.-matem. ob-va, Kazan 25, 86-113 (1926; JFM 52.0782.02)] and the Bochner technique (see \textit{H. Wu} [Proc. Beijing Symp. Diff. Geom. and Differ. Equat., Aug. 18 -- Sept. 21, 1980, New York 2, 929-1071 (1982; Zbl 0528.53042)]).
scientific article; zbMATH DE number 1135572
scientific article