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**Flat G-bundles with canonical metrics.**
*(English)*
Zbl 0676.58007

Let M be a Riemannian manifold and \(\pi\) : \(P\to M\) be a G-bundle over M, where G is a semisimple algebraic group over R. Let K be a maximal compact subgroup of G. A G-connection D on P decomposes as \(D=D^++\theta\), where \(D^+\) is a connection preserving the K- structure and \(\theta\) is a one form on M. The author defines \(\phi\) : \(D\to \phi_ D=D^{+,*}\) where the adjunction is taken with respect to the Killing form metric. He finds conditions under which a flat G- connection is mapped by some element of Aut(P) to a connection for which \(\phi\) is zero. Then he proves there exists a correspondence between the zeros of \(\phi\) and the stability of connections. As a consequence he obtains a classification of harmonic maps from a compact Riemannian manifold into a negatively curved locally symmetric manifold.

Further results on harmonic maps are given by using ideas of Y.-T. Siu [Ann. Math., II. Ser. 112, 73-112 (1980; Zbl 0517.53058) and Duke Math. J. 48, 857-871 (1981; Zbl 0496.32020)]. Finally he proves a conjecture of W. M. Goldman and J. Millson [Invent. Math. 88, 495-520 (1987; Zbl 0627.22012)].

Further results on harmonic maps are given by using ideas of Y.-T. Siu [Ann. Math., II. Ser. 112, 73-112 (1980; Zbl 0517.53058) and Duke Math. J. 48, 857-871 (1981; Zbl 0496.32020)]. Finally he proves a conjecture of W. M. Goldman and J. Millson [Invent. Math. 88, 495-520 (1987; Zbl 0627.22012)].

Reviewer: M.Anastasiei