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Register a new userBasic automorphism group of complete Cartan foliations covered by fibration
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We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the transverse Cartan geometry may not be effective. Some estimates of the dimension of this group depending on the transverse geometry are found. Further, we investigate Cartan foliations covered by fibrations. When the global holonomy group of that foliation is discrete, we obtain the explicit new formula for determining its basic automorphism Lie group. Examples of computing the full basic automorphism group of complete Cartan foliations are constructed.
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The paper is focused on the existence problem of attractors for foliations. Since the existence of an attractor is a transversal property of the foliation, it is natural to consider foliations admitting transversal geometric structures. As transversal structures are chosen Cartan geometries due to their universality. The existence problem of an attractor on a complete Cartan foliation is reduced to a similar problem for the action of its structure Lie group on a certain smooth manifold. In the case of a complete Cartan foliation with a structure subordinated to a transformation group, the problem is reduced to the level of the global holonomy group of this foliation. Each countable automorphism group preserving a Cartan geometry on a manifold and admitting an attractor is realized as the global holonomy group of some Cartan foliation with an attractor. Conditions on the linear holonomy group of a leaf of a reductive Cartan foliation sufficient for the existence of an attractor (and a global attractor) which is a minimal set are found. Various examples are considered.
We introduce a category of rigid geometries on singular spaces which are leaf spaces of foliations and are considered as leaf manifolds. We single out a special category $mathfrak F_0$ of leaf manifolds containing the orbifold category as a full subcategory. Objects of $mathfrak F_0$ may have non-Hausdorff topology unlike the orbifolds. The topology of some objects of $mathfrak F_0$ does not satisfy the separation axiom $T_0$. It is shown that for every ${mathcal N}in Ob(mathfrak F_0)$ a rigid geometry $zeta$ on $mathcal N$ admits a desingularization. Moreover, for every such $mathcal N$ we prove the existence and the uniqueness of a finite dimensional Lie group structure on the automorphism group $Aut(zeta)$ of the rigid geometry $zeta$ on $mathcal{N}$.
We present a new link between the Invariant Theory of infinitesimal singular Riemannian foliations and Jordan algebras. This, together with an inhomogeneous version of Weyls First Fundamental Theorems, provides a characterization of the recently discovered Clifford foliations in terms of basic polynomials. This link also yields new structural results about infinitesimal foliations, such as the existence of non-trivial symmetries.
We classify compact manifolds of dimension three equipped with a path structure and a fixed contact form (which we refer to as a strict path structure) under the hypothesis that their automorphism group is non-compact. We use a Cartan connection associated to the structure and show that its curvature is constant.
Any group $G$ gives rise to a 2-group of inner automorphisms, $mathrm{INN}(G)$. It is an old result by Segal that the nerve of this is the universal $G$-bundle. We discuss that, similarly, for every 2-group $G_{(2)}$ there is a 3-group $mathrm{INN}(G_{(2)})$ and a slightly smaller 3-group $mathrm{INN}_0(G_{(2)})$ of inner automorphisms. We describe these for $G_{(2)}$ any strict 2-group, discuss how $mathrm{INN}_0(G_{(2)})$ can be understood as arising from the mapping cone of the identity on $G_{(2)}$ and show that its underlying 2-groupoid structure fits into a short exact sequence $G_{(2)} to mathrm{INN}_0(G_{(2)}) to Sigma G_{(2)}$. As a consequence, $mathrm{INN}_0(G_{(2)})$ encodes the properties of the universal $G_{(2)}$ 2-bundle.
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