The topological matter of holonomy displacement on the principal U(n)-bundle over Dn,m, related to complex surfaces

JOURNAL OF GEOMETRY AND PHYSICS

卷:127

Article

Byun, Taechang¹ 

[1] Sejong Univ, Dept Math & Stat, Seoul 743747, South Korea

关键词: Holonomy displacement;    Area form;    Riemannian submersion;    Complex surface;    Complete totally geodesic submanifold;    Grassmannian manifold;   

DOI  :  10.1016/j.geomphys.2018.02.004

来源: Elsevier

【 摘 要 】

Consider U(n) -> U(n, m)/U(m) ->(pi) D-n,D-m, where D-n,D-m = U(n, m)/(U (n) x U(m)). Given a nontrivial X is an element of M-mxn(C) and g is an element of U(n, m), consider a complete oriented surface S = S(X, g) with a complex structure in D-n,D-m and a new area form omega((X,g)) on the surface S. Let c : [0, 1] -> S be a smooth, simple, closed, orientation-preserving curve and (c) over cap : [0, 1] -> U(n, m)/U(m) its horizontal lift. Then the holonomy displacement is given by the right action of e(psi) for some psi is an element of Span(R){i(X*X)(k)}(k=1)(p) subset of u(n), p = the number of distinct positive eigenvalues ofX*X, such that (c) over cap (1) = (c) over cap (0).e(psi) and Tr(psi) = 2i Area(c), where Area(c) is the area, produced by omega((X,g)), of the region on the surface S, surrounded by c. And psi can be represented as the solution of a system of first order ordinary linear differential equations. (C) 2018 Elsevier B.V. All rights reserved.

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