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**Metric rigidity theorems on Hermitian locally symmetric manifolds.**
*(English)*
Zbl 0912.32026

Series in Pure Mathematics. 6. Singapore: World Scientific. xiv, 278 p. (1989).

Hermitian symmetric spaces have long played a central role in classical complex differential geometry and they have been studied with various points of view in mind. The purpose of this monograph is to study the problem of characterizing canonical metrics on Hermitian locally symmetric spaces in terms of curvature conditions. As well, these metric rigidity theorems and their proofs are applied to the study of holomorphic maps between Hermitian locally symmetric spaces of the same type in order to get various rigidity theorems on holomorphic maps.

The presentation is divided into two parts. The first part contains a wealth of background material about complex differential geometry on Hermitian locally symmetric spaces, e.g. Hermitian connections, their curvatures, notions of positivity and negativity of curvature, explicit calculations of the curvature tensor of the Kähler form of the Bergman metric on classical bounded symmetric domains, etc. Also included is some of the author’s work on Hermitian metric rigidity in the case of compact quotients of bounded symmetric domains [N. Mok, Ann. Math., II. Ser. 125, No. 1, 105-152 (1987; Zbl 0616.53040)]. Suppose \(X\) is a compact quotient of an irreducible bounded symmetric domain of rank \(\geq 2\) and \(h\) is a Hermitian metric of (Griffiths) seminegative curvature. Then \(h\) is a constant multiple of the Kähler-Einstein metric \(g\). The monograph presents an expanded version of these results, e.g. description of and an integral formula on the characteristic bundle \(\mathcal S\), along with those in another paper by the author [Math. Ann. 276, No. 2, 177-204 (1987; Zbl 0612.53029].

The second part contains more recent developments. First, an extension of metric rigidity theorems to the case of finite volume due to W-K. To [Invent. Math. 95, No. 3, 559-578 (1989; Zbl 0668.53034)] is given. Next the immersion problem between complex hyperbolic space forms according to the author [“Local biholomorphisms between Hermitian locally symmetric spaces of noncompact type”, Preprint; per bibl.], and H. D. Cao and the author [Invent. Math. 100, No. 1, 49-61 (1990; Zbl 0698.53035)] is treated. Then follows a Hermitian metric rigidity theorem on locally homogeneous vector bundles.

The last chapter has a result of I.-H. Tsai [J. Differ. Geom. 33, No. 3, 717-729 (1991; Zbl 0718.53042)] on a rigidity theorem for holomorphic maps between irreducible Hermitian symmetric spaces of compact type. At the end there are four appendices dealing with some background material, e.g. on semisimple Lie algebras and their representations and on a dual generalized Frankel conjecture.

The presentation is divided into two parts. The first part contains a wealth of background material about complex differential geometry on Hermitian locally symmetric spaces, e.g. Hermitian connections, their curvatures, notions of positivity and negativity of curvature, explicit calculations of the curvature tensor of the Kähler form of the Bergman metric on classical bounded symmetric domains, etc. Also included is some of the author’s work on Hermitian metric rigidity in the case of compact quotients of bounded symmetric domains [N. Mok, Ann. Math., II. Ser. 125, No. 1, 105-152 (1987; Zbl 0616.53040)]. Suppose \(X\) is a compact quotient of an irreducible bounded symmetric domain of rank \(\geq 2\) and \(h\) is a Hermitian metric of (Griffiths) seminegative curvature. Then \(h\) is a constant multiple of the Kähler-Einstein metric \(g\). The monograph presents an expanded version of these results, e.g. description of and an integral formula on the characteristic bundle \(\mathcal S\), along with those in another paper by the author [Math. Ann. 276, No. 2, 177-204 (1987; Zbl 0612.53029].

The second part contains more recent developments. First, an extension of metric rigidity theorems to the case of finite volume due to W-K. To [Invent. Math. 95, No. 3, 559-578 (1989; Zbl 0668.53034)] is given. Next the immersion problem between complex hyperbolic space forms according to the author [“Local biholomorphisms between Hermitian locally symmetric spaces of noncompact type”, Preprint; per bibl.], and H. D. Cao and the author [Invent. Math. 100, No. 1, 49-61 (1990; Zbl 0698.53035)] is treated. Then follows a Hermitian metric rigidity theorem on locally homogeneous vector bundles.

The last chapter has a result of I.-H. Tsai [J. Differ. Geom. 33, No. 3, 717-729 (1991; Zbl 0718.53042)] on a rigidity theorem for holomorphic maps between irreducible Hermitian symmetric spaces of compact type. At the end there are four appendices dealing with some background material, e.g. on semisimple Lie algebras and their representations and on a dual generalized Frankel conjecture.

Reviewer: B.Gilligan (MR 92d:32046)

### MSC:

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |

32Q15 | Kähler manifolds |

32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |

32Q20 | Kähler-Einstein manifolds |

53C35 | Differential geometry of symmetric spaces |