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Infinitely Divisible Metrics, Curvature Inequalities And Curvature Formulae

Electronic Theses of Indian Institute of Science

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Title Infinitely Divisible Metrics, Curvature Inequalities And Curvature Formulae
 
Creator Keshari, Dinesh Kumar
 
Subject Hilbert Space
Curvature Inequalities
Cowen-Douglas Class Of Operators
Curvature of a Contraction
Jet Bundles (Mathematics)
Vector Bundles
Hermitian Holomorphic Vector Bundle
Infinitely Divisible Metrics
Kernel Functions
Geometry
 
Description The curvature of a contraction T in the Cowen-Douglas class is bounded above by the
curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples
of operators in the Cowen-Douglas class.
Secondly, we obtain an explicit formula for the curvature of the jet bundle of the Hermitian holomorphic bundle E f on a planar domain Ω. Here Ef is assumed to be a pull-back of the tautological bundle on gr(n, H ) by a nondegenerate holomorphic map f :Ω →Gr (n, H ).
Clearly, finding relationships amongs the complex geometric invariants inherent in the
short exact sequence
0 → Jk(Ef ) → Jk+1(Ef ) →J k+1(Ef )/ Jk(Ef ) → 0
is an important problem, whereJk(Ef ) represents the k-th order jet bundle. It is known that the Chern classes of these bundles must satisfy
c(Jk+1(Ef )) = c(Jk(Ef )) c(Jk+1(Ef )/ Jk(Ef )).
We obtain a refinement of this formula:
trace Idnxn ( KJk(Ef )) - trace Idnxn ( KJk-1(Ef ))= KJk(Ef )/ Jk-1(Ef )(z).
 
Contributor Misra, Gadadhar
 
Date 2014-06-30T06:22:43Z
2014-06-30T06:22:43Z
2014-06-30
2012-07
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/handle/2005/2332
http://etd.ncsi.iisc.ernet.in/abstracts/2999/G25296-Abs.pdf
 
Language en_US
 
Relation G25296