Local invariants of divergence-free webs. (English) Zbl 1509.53014
A divergence-free \(n\)-web is understood as a family \(\mathcal{W}=\{ \mathcal{F}_{1},\dots,\mathcal{F}_{n}\}\) of \(n\) foliations in general position on an \(m\)-dimensional manifold \(M\) endowed with a volume form \(\Omega\). A diffeomorphism \(\varphi :M\to N\) is an equivalence of divergence-free webs \(\{\mathcal{F}_{1},\dots,\mathcal{F}_{n}, \Omega_{M}\}\) on \(M\) and \(\{\mathcal{G}_{1},\dots,\mathcal{G}_{n}, \Omega_{N}\}\) on \(N\) if \(\varphi\) carries the leaves of \(\mathcal{F}_{j}\) onto the leaves of \(\mathcal{G}_{\sigma (j)}\), where \(\sigma\) is a permutation of \(\{ 1, \dots ,n\}\), and satisfies \(\varphi ^{*} \Omega_{N}=\Omega_{M}\). A divergence-free \(n\)-web is said to be locally trivial if it is locally equivalent to the trivial \(n\)-web of \(\mathbb{R}^{m}\).
In the present paper, two local invariants of such a geometric structure are introduced. The first one is related to the curvature of the natural connection associated to the structure. This connection was introduced by S. Tabachnikov [Differ. Geom. Appl. 3, No. 3, 265–284 (1993; Zbl 0789.53019)] for the case of 2-webs in a manifold with a volume form. The second invariant is the nonuniformity tensor of the divergence-free \(n\)-web, which is a symmetric covariant 2-tensor field \(\mathcal{K}\), recalling the curvature of the Chern connection of a 3-web (that measures if a 3-web is hexagonal or not), i.e., it is related to the notion of planar 3-web holonomy, studied many years before by W. Blaschke and G. Bol [Geometrie der Gewebe. Topologische Fragen der Differentialgeometrie. Berlin: Julius Springer (1938; JFM 64.0727.03)]. It is proved that the nonuniformity tensor of the divergence-free \(n\)-web of codimension 1 is the Ricci curvature tensor of the natural connection.
Main results of the paper concern characterizations of local triviality of such a divergence-free \(n\)-web. In particular, it is proved that vanishing of either of those invariants characterizes trivial divergence-free web-germs up to equivalence. The result corresponding to the first invariant generalizes another one obtained in the quoted paper of Tabachnikov for the case \(n=2\).
The paper is deep and it is carefully written. Besides, many examples are shown. The last section is about applications in general relativity.
In the present paper, two local invariants of such a geometric structure are introduced. The first one is related to the curvature of the natural connection associated to the structure. This connection was introduced by S. Tabachnikov [Differ. Geom. Appl. 3, No. 3, 265–284 (1993; Zbl 0789.53019)] for the case of 2-webs in a manifold with a volume form. The second invariant is the nonuniformity tensor of the divergence-free \(n\)-web, which is a symmetric covariant 2-tensor field \(\mathcal{K}\), recalling the curvature of the Chern connection of a 3-web (that measures if a 3-web is hexagonal or not), i.e., it is related to the notion of planar 3-web holonomy, studied many years before by W. Blaschke and G. Bol [Geometrie der Gewebe. Topologische Fragen der Differentialgeometrie. Berlin: Julius Springer (1938; JFM 64.0727.03)]. It is proved that the nonuniformity tensor of the divergence-free \(n\)-web of codimension 1 is the Ricci curvature tensor of the natural connection.
Main results of the paper concern characterizations of local triviality of such a divergence-free \(n\)-web. In particular, it is proved that vanishing of either of those invariants characterizes trivial divergence-free web-germs up to equivalence. The result corresponding to the first invariant generalizes another one obtained in the quoted paper of Tabachnikov for the case \(n=2\).
The paper is deep and it is carefully written. Besides, many examples are shown. The last section is about applications in general relativity.
Reviewer: Fernando Etayo Gordejuela (Santander)
MSC:
| 53A60 | Differential geometry of webs |
| 53A55 | Differential invariants (local theory), geometric objects |
| 53Z05 | Applications of differential geometry to physics |
| 57R05 | Triangulating |
| 58K50 | Normal forms on manifolds |
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