This article is basically the first chapter of the author’s PhD thesis. She explains in detail the Riemannian invariant called the \(K\)-area, introduced by M. Gromov [Positive curvature, macroscopic dimension, spectral gaps and higher signature, Functional Analysis on the eve of the 21st century, Vol II (New Brunswick, NJ, 1993), 1–213 (1996; Zbl 0945.53022)]. In the first section the author introduces different definitions of \(K\)-area. The \(K\)-area of a Riemannian manifold \((M;g)\), closed of even dimension, is the number \(K\text{-area} (M,g)=\frac{1}{\| R^X\|}\), where \(X\) is an Hermitian fibration not homologically trivial and \(\| R^X\|\) is an appropriate norm of the curvature. In fact the first definition does not work for open manifolds. The new definition admits that \(X\) is an Hermitian fibration trivial at infinity. A third definition is then introduced which allows manifolds of odd dimension.
Throughout the article the author gives detailed proofs of some properties of the \(K\)-area already present in some of M. Gromov’s work [loc. cit. and Structures métriques pour les variétés riemanniennes. Paris: Cedic (1981; Zbl 0509.53034)] and joint work with H. B. Lawson jun. [Ann. Math. (2) 111, 209–230 (1980; Zbl 0445.53025), Ann. Math. (2) 111, 423–434 (1980; Zbl 0463.53025), Publ. Math., Inst. Hautes Étud. Sci. 58, 83–196 (1983; Zbl 0538.53047)]. Each property is illustrated by examples, either in projective complex spaces, canonical spheres, Euclidean balls or Euclidean spaces. These properties include: the fact that the \(K\)-area is positive; relations between the \(K\)-area of different manifolds related by special maps. A special section is dedicated to the behaviour of the \(K\)-area of connected sums.
The last part of the article is dedicated to the fundamental theorem introduced in M. Gromov [loc. cit., Zbl 0945.53022], that ties the \(K\)-area to the scalar curvature for spin manifolds. This theorem gives new obstruction to the existence of a metric of strictly positive curvature in the same sense as the enlargement introduced in [loc. cit., Zbl 0463.53025]. The very last issue of the article is the statement of a generalisation of this theorem, proven in the authors thesis.
For the entire collection see [Zbl 1032.35005].