Geometry on Manifolds: Riemannian Metrics, Symplectic Forms, Contact Structures (Compact Textbooks in Mathematics)

This first course on differential geometry includes not only Riemannian manifolds, but also symplectic and contact geometry.Rather than treating these three fields of geometry as separate subjects, this text emphasises common features by organising the material according to ideas and methods shared by the three fields. Specifically, this text highlights how certain concepts, such as structure-preserving vector fields or variational characterisations of curves adapted to a given geometric structure, find their analogous expressions in the respective field of geometry.For example, Frobenius integrability, which is primarily relevant for contact geometry, is discussed together with the classification of flat Riemannian manifolds, which requires very similar arguments. Another case in point is the discussion of transformation groups (isometries, symplectomorphisms, contactomorphisms) and the corresponding Lie algebras.Two equivalent ways to construct this Lie algebra structure are describe