A Riemannian manifold \((V,g)\) is called Kählerian if \(g\) can be completed to a Kähler structure \((g,\omega ,J)\) with suitable \(\omega \) and \(J\). The main purpose of this paper is to study metric invariants of \((V,g)\) which take special values on Kählerian manifolds. Besides finding a metric and geometric characterization of Kähler manifolds, the author studies the measure of deviation (or distance) of an arbitrary Riemannian manifold from the Kählerian locus in the space of all Riemannian manifolds. Using these invariants, one can see how much non-Kählerian they may become on general Riemannian manifolds.
Among the invariants which appear in this paper are the ones relating to the volumes of submanifolds in \((V,g)\) of positive codimension. Following M. Berger, \(\text{sys}_k(V,g)\) is the infimum of the volumes of \(k\)-dimensional submanifolds (or more general cycles) in \(V\) which are not homologous to zero. Their evaluation for some complex algebraic manifolds is given. To show that the range of the systolic invariants drastically increases when one passes from Kähler to all Riemannian manifolds, the author gives the corresponding examples.
The second group of invariants refers to continuous maps \(\varphi :V\rightarrow W\) where \(W=(W,h)\) is a standard Riemannian manifold (e.g., a flat Riemannian torus \(T^d\)) and where \(\varphi \) belongs to a suitable homotopy class \(\Phi \) of maps \(V\rightarrow W\). Then \(\text{Min En } V=\text{Min En} (V|W,\Phi )\) as the infimum of the Dirichlet energies of the maps \(\varphi \) in \(\Phi \). If \(V\) is Kähler, then the author presents holomorphic maps \(\varphi \) which minimize the energy and, moreover, equate \(\text{Min En } V\) to some purely topological invariant of \(V\). He also indicates non-Kähler examples where Min En may become arbitrarily large compared to the Kähler case.
The last group of invariants refers to the spectrum \(\{ 0<\lambda _1\leq \lambda _2 \leq \dots \leq \lambda _k\leq \dots \} \) of the Laplace operator on \(V\). A non-trivial upper bound for all eigenvalues \(\lambda _k\) is given in the Kähler case. The author also recalls examples showing that there are no restrictions on individual \(\lambda _k\) for general Riemannian manifolds. Some open problems and conjectures are also given.
For the entire collection see [Zbl 0879.00051].