By Franki J.E. Dillen, Leopold C.A. Verstraelen

Within the sequence of volumes which jointly will represent the guide of Differential Geometry a slightly entire survey of the sector of differential geometry is given. different chapters will either care for the elemental fabric of differential geometry and with study effects (old and recent). All chapters are written by way of specialists within the region and comprise a wide bibliography.

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4. Riemannian foliations . . . . . . . 5. Transversely holomorphic foliations . . . 3. Codimension one foliations . . . . . . . 4. Γ -structures . . . . . . . . . . . 5. The leaf space . . . . . . . . . . . 1. V -manifolds . . . . . . . . . . 2. QF -manifolds . . . . . . . . . 3. Q-manifolds . . . . . . . . . . 6. Characteristic classes . . . . . . . . . 1. 2. Classification of real vector bundles . . . 7. Basic global analysis .

64 64 65 66 Foliations 37 0. Foreword Foliation Theory is the qualitative study of Differential Equations. It was initiated by the works of H. Poincaré, I. Bendixon and developed later by C. Ehresmann, G. Reeb and many other people. Since then the subject has been a wide field in mathematical research. Actually it is almost impossible to describe all the results and the different steps of its development.

The integral curves of X are geodesics parameterized with unit speed) defined on a neighborhood O of m and such that X(m) = vm . If we define a Riemannian metric on O by x → gϕ (X(x)) as in Section 2, then the Riemannian curvature of this metric at m equals the Finsler curvature of (M, ϕ) at vm . 4. Cartan’s structure equations By now we have defined three geometric invariants of Finsler surfaces: I , J and K. The invariant I is a centro-affine invariant which describes the shape of each unit tangent circle, the invariant K belongs to the calculus of variations and measures the focusing of geodesics, and the invariant J , by measuring how I varies along geodesics, partakes of both convex geometry and variational calculus.