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On an axiomatic treatment of differential geometry via vector sheaves. Applications. (English) Zbl 0910.53013

The geometry of differential manifolds and fiber bundles is, undoubtedly, the core of a very rich theory interacting with many fields of mathematics and numerous applications. However, the theory of smooth manifolds (based on ordinary differential calculus) is inadequate to handle spaces like orbifolds, spaces with corners or singularities, etc. These deficiencies led to the study of, generally speaking, various differential spaces in which new classes of smooth functions, enlarging the ordinary ones, are considered.
Circa 1998, the author of the present paper, starting from a completely different point of view, had the idea to replace the functional structure sheaf of the previous spaces (the classical case, of course, included) by an arbitrary sheaf \({\mathcal A}\) of unital, commutative and associative \(\mathbb{C}\)-algebras over \(X\), thus arriving at the notion of a \(\mathbb{C}\)-algebraized space \((X, {\mathcal A})\). He further assumed that to \((X,{\mathcal A})\) there is associated a differential triad \(({\mathcal A}, \partial, \Omega)\), where \(\Omega\) is an \({\mathcal A}\)-module over \(X\) and \(\partial: {\mathcal A} \to \Omega\) a \(\mathbb{C}\)-linear morphism satisfying the Leibniz condition \(\partial (s\cdot t) =s\cdot \partial(t) +t\cdot \partial (s)\) for any local sections \(s,t\in {\mathcal A} (U)\) and \(U \subseteq X\) open.
It should be noted that, although the terminology applied is reminiscent of the classical theory, there is not any kind of differentiability involved, the structure of such a triad being in fact ensured by Kähler’s (algebraic) theory of differentials.
The previous considerations constitute the basis of an abstract differential geometry developed by the author, whose research program in this direction has culminated in his recent two-volume book [A. Mallios, ‘Geometry of vector sheaves. An axiomatic approach to differential geometry’ (Math. Appl. 439, Kluwer, Dordrecht) (1998; Zbl 0904.18001, Zbl 0904.18002)]. The paper under review is essentially an extensive summary (of approx. 90 pages!) of the previous work, complemented also by three appendices devoted to applications referring to geometric prequantization, Yang-Mills theory and Gribov’s ambiguity, not included in the book.
The most important objects of this study are, of course, vector sheaves and their connections. By the former we mean \({\mathcal A}\)-modules \({\mathcal E}\) over \(X\), which locally look like \({\mathcal A}^n\) (if the rank of \({\mathcal E}\) is \(n)\), whereas a connection on \({\mathcal E}\) is a \(\mathbb{C}\)-linear morphism \(D:{\mathcal E} \to\Omega \otimes_{\mathcal A} {\mathcal E}\) such that \(D(\alpha\cdot s)=\alpha\cdot D(s)+s\otimes\partial(\alpha)\), for any \(\alpha \in {\mathcal A} (U)\), \(s\in {\mathcal E}(U)\) and \(U\subseteq X\) open. The theory build there upon is the abstraction of the theory of connections on ordinary vector bundles and comprises the analogs of such “classical” themes as existence theorems, induced connections, local connection forms. Christoffel symbols, Riemannian and Hermitian connections, as well as the various curvatures, Cartan’s structural equation, Bianchi’s identity, flat connections, Chern classes, Weil’s integrality theorem, classification of Maxwell fields (i.e., line bundles endowed with connections), and many others.
It is quite surprising that a great deal of the standard differential geometry can be obtained in this abstract framework without any differentiability, by applying only algebro-topological methods within the theory of sheaves and sheaf-cohomology. Of course, there are certain cases (e.g., Bianchi’s identity, the equivalence of various notions of flatness – based on an abstract Frobenius integrability condition – Weil’s integrality theorem, classification of Maxwell fields) in which some further axioms should be added. This is quite natural if we take into account the generality of the present framework and the few tools available, in contrast to the abundance of means existing in the classical case, due to the flexibility and richness of the calculus. In this respect, at certain stages one should assume (separately or in combinations) the exactness of the generalized de Rham complex \[ 0\to \mathbb{C}\to {\mathcal A} @>\partial>> \Omega \equiv \Omega^1 @>d^1>> \Omega^2 \to\cdots \to \Omega^n @>d^n>> \cdots \] (where \(\Omega^n: =\wedge^n \Omega\) and the \(d^i\)’s are appropriate extensions of \(\partial)\), the exactness of the exponential sheaf sequence \[ 0\to \mathbb{Z} @>i>> {\mathcal A} @>e>> {\mathcal A}^\bullet \to 1 \] \(({\mathcal A}^\bullet\) is the group sheaf of invertible elements of \({\mathcal A}\) and \(e\) an appropriate exponential morphism), as well as the equality \[ {1\over 2\pi i} \widetilde \partial \circ e= \partial, \] \(\widetilde \partial\) being the logarithmic differential given by \(\widetilde \partial(s): =s^{-1} \cdot \partial (s)\), for any \(s\in {\mathcal A}^\bullet (U) \cong {\mathcal A} (U)^\bullet\) and \(U\subseteq X\) open. However, suitable topological algebra sheaves provide conditions under which the previous assumptions are fulfilled.
The paper also contains two chapters dealing with the basics of the theory of sheaves, cohomology and the algebra of vector sheaves, one chapter pertaining to various examples and applications and the three appendices mentioned earlier. There are no proofs. The wealth of notions, statements and comments is considerable. The exposition not only explains what is going on in this new approach but also clarifies all subtleties appearing in the transition from the classical theory towards this abstraction.
In conclusion, the paper is a very information introduction to the subject in the title, after the study of which the interested geometer will certainly shift for details to the aforementioned book by the same author.
Reviewer’s remarks. The geometry of principal sheaves (the analog of principal bundles within this abstract context) is the aim of a research program by the reviewer; see, e.g., his article [E. Vassiliou, ‘Connections on principal sheaves’, in: New Developments in Differential Geometry (Budapest 1996), 459-483 (1999)] and the references of the present paper.
The reviewer understands that A. Mallios is completing a third volume, dealing with gauge theories and variational problems in the same spirit.

MSC:

53C05 Connections (general theory)
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
57R20 Characteristic classes and numbers in differential topology
58A03 Topos-theoretic approach to differentiable manifolds
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
58D27 Moduli problems for differential geometric structures
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
55N30 Sheaf cohomology in algebraic topology
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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