Metric Rigidity Theorems on Hermitian Locally Symmetric by Ngaiming Mok

By Ngaiming Mok

This monograph stories the matter of characterizing canonical metrics on Hermitian in the neighborhood symmetric manifolds X of non-compact/compact varieties when it comes to curvature stipulations. The proofs of those metric pressure theorems are utilized to the research of holomorphic mappings among manifolds X of an identical kind. in addition, a twin model of the generalized Frankel Conjecture on characterizing compact Kähler manifolds also are formulated.

Show description

Read Online or Download Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds (Series in Pure Mathematics, V. 6) PDF

Best topology books


Even though the quest for strong selectors dates again to the early 20th century, selectors play an more and more vital position in present study. This publication is the 1st to gather the scattered literature right into a coherent and stylish presentation of what's recognized and confirmed approximately selectors--and what is still discovered.

From Topology to Computation: Proceedings of the Smalefest

A rare mathematical convention was once held 5-9 August 1990 on the collage of California at Berkeley: From Topology to Computation: harmony and variety within the Mathematical Sciences a global learn convention in Honor of Stephen Smale's sixtieth Birthday the themes of the convention have been a few of the fields during which Smale has labored: • Differential Topology • Mathematical Economics • Dynamical platforms • conception of Computation • Nonlinear practical research • actual and organic purposes This booklet contains the court cases of that convention.

Applications of Contact Geometry and Topology in Physics

Even supposing touch geometry and topology is in short mentioned in V I Arnol'd's booklet "Mathematical tools of Classical Mechanics "(Springer-Verlag, 1989, 2d edition), it nonetheless is still a website of study in natural arithmetic, e. g. see the new monograph by means of H Geiges "An advent to touch Topology" (Cambridge U Press, 2008).

Why Prove it Again?: Alternative Proofs in Mathematical Practice

This monograph considers numerous famous mathematical theorems and asks the query, “Why turn out it back? ” whereas analyzing replacement proofs. It explores the several rationales mathematicians could have for pursuing and proposing new proofs of formerly tested effects, in addition to how they pass judgement on even if proofs of a given end result are assorted.

Extra resources for Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds (Series in Pure Mathematics, V. 6)

Sample text

If A > 0, then (X,g) is non—compact and semisimple. In the latter two cases we say that (X,g) is an irreducible Riemannian symmetric manifold of compact type and non—compact type resp. In general when G is semisimple we say that (X,g) is of semisimple type. Let g be a semisimple Lie algebra, be a compact subalgebra and 0 be an involution on g such that is the fixed point set of 0. We call a semisimple orthogonal symmetric Lie algebra. As described above, given any simplyconnected Riemannian symmetric manifold (X,g) without flat factors there corresponds such a pair Conversely, given such a pair (g,r,0) one can reconstruct a simply-connected Riemannian symmetric manifold (X,g) by taking G to be a simply—connected Lie group with Lie algebra g, K c G to be a connected subgroup corresponding to the Lie subalgebra C g, and the Riemannian metric g to be induced by a right—invariant metric on G.

By the uniqueness of the decomposition of tangent vectors into components of types (1,0) and (0,1) it follows that along 'y. In other words, and and are both parallel along (1) Suppose (2) Since J,7 = — it follows that is parallel along (2) (1) Let be a tangent vector of type (1,0) at 'y(o) and write for its parallel transport along Let be the decomposition into + = = — components of types (1,0) and (0,1). By hypothesis (2) is and are parallel along Since also parallel. It follows that both vanishes identically on 7, so that the parallel = 0 we conclude that transport of proving (2) remains to of type (1,0) along (1).

Over x E isomorphic We are going to compute the curvature of (L$). Let x E X and be a system of local holomorphic coordinates at x. Let (va)i(a(r be special holomorphic fiber coordinates for V at x. Denote by the corresponding holomorphic basis of V over a neighborhood of x. ,Zn yr F(V) E ; (and where 0). For the Hermitian holomorphic line bundle (L,h) over P(V) we use the holomorphic fiber coordinate A = in a neighborhood of We assert PROPOSITION 1 In terms of the coordinates given above A is a special holomorphic fiber coordinates for L adapted to h at [p) and, denoting the curvature form of (V,h) by e, we have = -41 duU A diia + dz' A As a consequence (L$) is of seminegative curvature if and only if (V,h) is of seminegative curvature.

Download PDF sample

Rated 4.68 of 5 – based on 43 votes