On the diffeomorphism groups of foliated manifolds. (English. Russian original) Zbl 1544.53022

Narmanov, A. Ya.; Sharipov, A. S.

J. Math. Sci., New York 276, No. 6, 767-775 (2023); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 181, 74-83 (2020).

Assume that \(M\) is a manifold and \(\mathrm{Diff}(M)\) is the set of all diffeomorphisms of \(M\) onto itself. We know that \(\mathrm{Diff}(M)\) is a group with respect to the composition and inverting mappings.
If \(M\) is a finite-dimensional manifold, then the isometry group \(I(M)\) of the Riemannian manifold \(M\) is a Lie group. Let \(M\) be a smooth connected Riemannian manifold of dimension \(n\), \(0< k <n\). A foliation \(F\) of dimension \(k\) (or a foliation of codimension \(n-k\)) is a partition of \(M\) into linearly connected subsets \(L_\alpha\subset M\) satisfying the following properties:

(\(F_1\))

\(\bigcup_{\alpha\in B}L_\alpha=M\);

(\(F_2\))

if \(\alpha\neq \beta\), then \(L_\alpha\cap L_\beta=\emptyset\) for all \(\alpha, \beta\in B\);

(\(F_3\))

for any point \(p\in M\), there exist a neighborhood \(U\) and a chart \((x^1,\dots,x^k,y^1,\dots,y^{n-k})\) such that for each leaf \(L_\alpha\), the linear connected components of the set \(U_p\cap L_\alpha\) are determined by the equations \(y^1=\mathrm{const},\cdots,y^{n-k}=\mathrm{const}\).

Now, by \((M, F)\) denote a smooth manifold \(M\) of dimension \(n\), \(0<k<n\), on which a smooth \(k\)-dimensional foliation \(F\) is defined. We denote by \(\mathrm{Diff}_F(M)\) the set of all \(C^r\)-diffeomorphisms of the foliated manifold \((M, F)\) for some fixed \(r\ge 0\). The set \(\mathrm{Diff}_F(M)\) is a group with respect to the operations of composition and inverting mappings. Note that \(\mathrm{Diff}_F(M) \le \mathrm{Diff}(M)\). The authors prove that \(\mathrm{Diff}_F(M)\) is a topological group with the \(F\)-compact-open topology.
The authors consider some examples of foliated manifolds and examine some subgroups of the group of diffeomorphisms of the foliated manifold.

Reviewer: Ali Morassaei (Zanjan)

MSC:

53C12	Foliations (differential geometric aspects)
57R30	Foliations in differential topology; geometric theory
57R50	Differential topological aspects of diffeomorphisms
57S05	Topological properties of groups of homeomorphisms or diffeomorphisms
58D05	Groups of diffeomorphisms and homeomorphisms as manifolds

Keywords:

foliations; compact-open topology; diffeomorphism groups

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