Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Nag, Subhashis

Abstract (Summary of the Work)

This thesis ie concerned chiefly with oertain families of smooth ... mappinga from domains in IRn to IRn which arise as natural generali- setione of complex anelytio mappinge. Thess Fueter families heve important hypercomplex aubfamilies defined via convergent Laurent series in one queternionio or actonionio veriable (with central coeffi- ofents). The various families and subfamdlies are olosed under compositior and inversion (when defined) of, meppinge consequently they form pseudo- groups on which one can model manifolds. The geometry of manifolde which oerry such ;Fueter or hypercomplex structure turna out to be qui ta %3D rich. In particular, they heve oanonioal foliations and the leaves of foliatione carry naturel complex mandfold atructures. Further, such manifolds coma equipped with a fibre-bundle projection to spheres of appropriate dimenedon. These properties allow ue to inveetigate the topological and analytio atructure of Fueter and hypercomplex manifolde ueing the methods of algebraic topology and function theory. For example,we prove in Chapter IV, using fairly subtle topology, that the only Dompact simply connected manifolds which can possibly allow queternionic (rospeotivoly ootonionie) structures ere s (reepectively s) (theso are the quaterniondo and actonionic projective spaces) and s2 x s2 (rospectively s² x s6).In order to study these manifolda equipped with Fueter and/or hypercomplex ocoor dinate systema it ie of course imperative to firet understand the geometric and analytic nature of the Fueter and hyper- complex mappinge themselvee. This study is carried out in Cheptore II and III. Indeoed, we find that the hypercomp lex mappings satisfy certe generalised Cauahy-Riomann relations, We are able to aupploment the Cauchy-Riemenn relatione uith some extra algebro-diPferential identita satiefied by our mappings. Thus ue can characterise Fueter and hyper- complex mappinge by a fixed ayetem of partial differential aquations. We determine conditions for the K-quesioonformality of our mapping in the seneo of Ahlfore [1].The problem of whether a given c menifold cen be aseignad hyperçomp lex/Fueter structurs ie of course intinately related to wheth the atruoture group of the tangent bundle of the manifold can be reduo to the group of Jacobiane of, hypercomplex/Fueter diffeamorphiome.


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