## Riemannian geometry of foliations

EU-funded mathematicians have developed new concepts and tools for the study of foliations, the extrinsic geometric flows introduced as deformations of metrics on foliated manifolds.

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Geometric flow is called the evolution of a geometric structure on a manifold under a differential equation, usually associated with curvature. This popular mathematical concept describes the metric of a Riemannian manifold deforming in an analogous manner to heat diffusion in physics while irregularities are smoothed.

If this evolution depends on the second fundamental tensor of the foliation, the geometric flow on a foliated manifold is called extrinsic. Within the EU-funded project EGFLOW (Extrinsic geometric flows on foliated manifolds), mathematicians introduced extrinsic geometric flow to the study of the codimension one foliations.

The EGFLOW team studied the geometry of a codimension-one foliation with a time-dependent Riemannian metric. They started with deformations of geometric quantities observed as the Riemannian metric changes along the leaves of the manifold foliation.

Next, mathematicians looked into the geometric flow of the so-called second fundamental form of the foliation. The evolution of this extrinsic curvature yielded, under certain assumptions, the second parabolic partial differential equations for which the existence and uniqueness of a solution were also shown.

Furthermore, they explored applications of the extrinsic geometric flow to the minimisation of the Einstein-Hilbert action, involving the scalar curvature being replaced by the mixed scalar curvature. The action was studied for codimension one foliation on a closed Riemannian manifold.

The parabolic geometric flow was extended to cover arbitrary codimension foliations, where the speed of the flow is proportional to the divergence of the mean curvature vector. The flow is equivalent to the heat flow of one-form, namely the dual of the curvature vector field, and admits a unique global solution, converging to a metric.

Part of EGFLOW was devoted to the Ricci flow, another popular geometric flow in mathematics. The partial Ricci flow was also introduced for codimension one foliation as well as for arbitrary codimension foliations. Specifically, an extrinsic representation of the partial Ricci flow on rotational surfaces was developed.

Geometric flow equations are difficult to solve because of their nonlinearity. EGFLOW was, therefore, centred around special cases. Many results were obtained that are of interest to geometers specialising in Riemannian geometry and foliations.

If this evolution depends on the second fundamental tensor of the foliation, the geometric flow on a foliated manifold is called extrinsic. Within the EU-funded project EGFLOW (Extrinsic geometric flows on foliated manifolds), mathematicians introduced extrinsic geometric flow to the study of the codimension one foliations.

The EGFLOW team studied the geometry of a codimension-one foliation with a time-dependent Riemannian metric. They started with deformations of geometric quantities observed as the Riemannian metric changes along the leaves of the manifold foliation.

Next, mathematicians looked into the geometric flow of the so-called second fundamental form of the foliation. The evolution of this extrinsic curvature yielded, under certain assumptions, the second parabolic partial differential equations for which the existence and uniqueness of a solution were also shown.

Furthermore, they explored applications of the extrinsic geometric flow to the minimisation of the Einstein-Hilbert action, involving the scalar curvature being replaced by the mixed scalar curvature. The action was studied for codimension one foliation on a closed Riemannian manifold.

The parabolic geometric flow was extended to cover arbitrary codimension foliations, where the speed of the flow is proportional to the divergence of the mean curvature vector. The flow is equivalent to the heat flow of one-form, namely the dual of the curvature vector field, and admits a unique global solution, converging to a metric.

Part of EGFLOW was devoted to the Ricci flow, another popular geometric flow in mathematics. The partial Ricci flow was also introduced for codimension one foliation as well as for arbitrary codimension foliations. Specifically, an extrinsic representation of the partial Ricci flow on rotational surfaces was developed.

Geometric flow equations are difficult to solve because of their nonlinearity. EGFLOW was, therefore, centred around special cases. Many results were obtained that are of interest to geometers specialising in Riemannian geometry and foliations.