EXTGEOMFOL

Integral formulae and extrinsic geometry of foliations

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 Obiettivo del progetto (Objective)

'The project concerns investigations of the extrinsic geometry (expressed by the 2nd fundamental form of leaves) and its relations with mixed curvature, topology and dynamics of foliations on compact (or of finite volume) manifolds. The research objectives are to go deeply in studying foliations using the methods of Riemannian geometry, theory of submanifolds, topology and dynamics, computer simulations. Among the (hoped for) results are new integral formulae for foliations on Riemannian manifolds, connectedness properties of (k, ε)-saddle (saddle, totally geodesic) foliations, and applications of them to the following problems: – Walczak problem: to recover a metric on a foliated manifold by higher mean curvatures of a foliation or its orthogonal distribution; – Gluck-Ziller problem: minimizing certain functions like volume and energy defined for plane fields on Riemannian manifolds, considered as maps into the (co)tangent Grassmann bundles equipped with Cheeger-Gromoll type metrics; – Toponogov problem: to generalize Ferus’s theorem for the case of foliations with non-negative (positive) mixed curvature. The topic belongs to differential geometry and topology, subjects of pure mathematics.'

Introduzione (Teaser)

An EU-funded study developed new methods for solving complex geometric problems dealing with entirely abstract concepts.

Descrizione progetto (Article)

Geometry is all about the shape and the properties of mathematical objects, while topology is mainly concerned with the properties that remain unchanged after the continuous deformation of such objects. The notion of a manifold captures the idea of a curved space that simply looks flat in small local regions. These regions or sub-manifolds are geometric devices for analysing manifolds called foliations or leaves.

The EU-funded research project 'Integral formulae and extrinsic geometry of foliations' (Extgeomfol) studied the geometric properties of the foliations of such objects and their relationships with topology. It investigated these properties in depth using geometric analysis methods, topological theories and computer simulations.

The main results were the development of three novel research tools: extrinsic geometric flow (EGF), integral formulae (IF) and variation formulae (VF) as well as their applications to solving mathematical problems on foliations.

The IF are useful for solving complex geometric problems concerning suitable objects. This includes defining their higher mean curvatures and minimising their volume and energy as defined for their vector or plane fields. However, the important result is EGF, which is a tool for defining the geometric properties of certain objects. To use this it is necessary to use the VF, which represent a change in the different geometric quantities of these objects under set conditions.

In the past, applications for pure mathematical problems have included sending information securely using encryption techniques and other means of 'coding' and 'decoding' for checking and correcting errors in signal processing and data transmissions. The project's technical achievements open the way to generating new knowledge in subjects relevant to the field of pure mathematics.