Final Report Summary - EXTGEOMFOL (Integral formulae and extrinsic geometry of foliations)
The project objectives are to study extrinsic geometry of foliations (properties expressed by the second fundamental form b of leaves) and its relations with topology / dynamics. Main results concern the development of research tools:
1) Extrinsic geometric flow (EGF)
2) Integral formulae (IF)
3) Variation formulae (VF)
and applications of them to problems on foliations. The research could be relevant to geometers and young specialists in areas of Riemannian geometry and foliations, in Europe, the United States of America (USA), Japan, etc.
1. EGF is defined as deformations of metrics on a manifold equipped with a codimension-1 foliation subject to quantities expressed in terms of b. It is a tool for studying the question: Under what conditions on (M, F, g(0)) the metrics g(t) converge to one for which F is totally umbilical or minimal.
Definition: Let (M, F) be a foliated manifold equipped with a transversal vector field N. Given transformation h acting on F-truncated symmetric (0, 2)-tensors, a family g(t) of Riemannian metrics on M making N unit normal to F and satisfying dg/dt = h(g) is called EGF.
Let h(b) = sum_(j < n) f(j) bj, where bj are (0, 2)-tensors dual to j-th degree of Weingarten operator A, extended on TM by A(N) = 0. Auxiliary functions f(j) depend on t(i), I = 1...n - the power sums of principal curvatures of F. The terms bj are meaningful: the Newton transformation flow depends on all bj, j < i.
Definition: EG soliton (EGS) on a foliated manifold (M, F, N) is a pair (g, X) of a metric, and a complete vector field preserving F satisfying h(b) = e g + L_X g (with the Lie derivative) for some real e. Depending on X, EGS might be tangent and normal.
The existing / classifying EGS have great interest and are related to the basic problem in theory of foliations (first posed by Gluck, 1979, for geodesic foliations).
Problem: Given (P), a property of Riemannian submanifolds expressed in terms of b, and a foliated manifold (M, F), decide if there exist (P)-metrics on (M, F), that is, all the leaves enjoy the property (P). If they do exist, study their properties, classify them.
EGF is really original approach which should provide striking new strategies and result in studying the extrinsic geometry of foliations. The approach is somehow similar to that for the Ricci flow which implied solutions of Poincare conjecture and Thurtston geometrisation conjecture. The EGF extends the application area of IF's and VF's for foliations.
The results are the local existence / uniqueness theorems, estimating the existence time of solutions, the convergeness in a sense to minimal and totally geodesic foliations, geometry of extrinsic Ricci and Newton transformation flows, classifying EGS among totally umbilical foliations and foliated surfaces.
2. If for a foliation means the vanishing of a total over M expression composed from quantities depending on b (as higher mean curvatures), the co-nullity, integrability and the curvature tensor. The results generalise Walczak (1990), Andrzejewski and Walczak (2010) (for special functions fj providing Newton transformations of the co-nullity tensor); and Brito, Langevin and Rosenberg (2001, with constant curvature) formulae.
3. We obtain VF when either 3a) foliation or 3b) metric are varied.
3a) The results are VF for the i-th mean curvature: if M has constant curvature they do not depend on the choice of k vector fields (Brito, Langevin and Rosenberg, 1981, for k=1). We prescribe the partial Ricci curvature of a foliation: find conditions for a symmetric 2-tensor to admit conformal metrics, solve partial Ricci equations, compute variations of the total mixed curvature. We deduce VF for the total mixed scalar curvature, and apply them to the to total energy/bending.
3b) The results are VF for total quantities under F-truncated variations g(t), geometry of critical metrics and applications to EGF related functionals. Combining with IF approach we find VF for the mixed scalar curvature and energy / bending.
1) Extrinsic geometric flow (EGF)
2) Integral formulae (IF)
3) Variation formulae (VF)
and applications of them to problems on foliations. The research could be relevant to geometers and young specialists in areas of Riemannian geometry and foliations, in Europe, the United States of America (USA), Japan, etc.
1. EGF is defined as deformations of metrics on a manifold equipped with a codimension-1 foliation subject to quantities expressed in terms of b. It is a tool for studying the question: Under what conditions on (M, F, g(0)) the metrics g(t) converge to one for which F is totally umbilical or minimal.
Definition: Let (M, F) be a foliated manifold equipped with a transversal vector field N. Given transformation h acting on F-truncated symmetric (0, 2)-tensors, a family g(t) of Riemannian metrics on M making N unit normal to F and satisfying dg/dt = h(g) is called EGF.
Let h(b) = sum_(j < n) f(j) bj, where bj are (0, 2)-tensors dual to j-th degree of Weingarten operator A, extended on TM by A(N) = 0. Auxiliary functions f(j) depend on t(i), I = 1...n - the power sums of principal curvatures of F. The terms bj are meaningful: the Newton transformation flow depends on all bj, j < i.
Definition: EG soliton (EGS) on a foliated manifold (M, F, N) is a pair (g, X) of a metric, and a complete vector field preserving F satisfying h(b) = e g + L_X g (with the Lie derivative) for some real e. Depending on X, EGS might be tangent and normal.
The existing / classifying EGS have great interest and are related to the basic problem in theory of foliations (first posed by Gluck, 1979, for geodesic foliations).
Problem: Given (P), a property of Riemannian submanifolds expressed in terms of b, and a foliated manifold (M, F), decide if there exist (P)-metrics on (M, F), that is, all the leaves enjoy the property (P). If they do exist, study their properties, classify them.
EGF is really original approach which should provide striking new strategies and result in studying the extrinsic geometry of foliations. The approach is somehow similar to that for the Ricci flow which implied solutions of Poincare conjecture and Thurtston geometrisation conjecture. The EGF extends the application area of IF's and VF's for foliations.
The results are the local existence / uniqueness theorems, estimating the existence time of solutions, the convergeness in a sense to minimal and totally geodesic foliations, geometry of extrinsic Ricci and Newton transformation flows, classifying EGS among totally umbilical foliations and foliated surfaces.
2. If for a foliation means the vanishing of a total over M expression composed from quantities depending on b (as higher mean curvatures), the co-nullity, integrability and the curvature tensor. The results generalise Walczak (1990), Andrzejewski and Walczak (2010) (for special functions fj providing Newton transformations of the co-nullity tensor); and Brito, Langevin and Rosenberg (2001, with constant curvature) formulae.
3. We obtain VF when either 3a) foliation or 3b) metric are varied.
3a) The results are VF for the i-th mean curvature: if M has constant curvature they do not depend on the choice of k vector fields (Brito, Langevin and Rosenberg, 1981, for k=1). We prescribe the partial Ricci curvature of a foliation: find conditions for a symmetric 2-tensor to admit conformal metrics, solve partial Ricci equations, compute variations of the total mixed curvature. We deduce VF for the total mixed scalar curvature, and apply them to the to total energy/bending.
3b) The results are VF for total quantities under F-truncated variations g(t), geometry of critical metrics and applications to EGF related functionals. Combining with IF approach we find VF for the mixed scalar curvature and energy / bending.