Review by Choice Review
In a certain modern usage, "geometry" and "topology" signal opposite tendencies in the study of structured mathematical spaces. Generally, one has geometry when one can "feel" variations of structure when moving from point to point; one has topology when one can only make global distinctions because nothing local distinguishes one point from another. Geometry often features numerical or metric structures (length, angle, curvature) rendering spaces rigid, while topology tends toward purely set-theoretic structures leaving spaces flexible. Certain classical structures, although given by numerical information (special sorts of differential forms), nevertheless show a topological character for nontrivial reasons (Darboux's theorems). In particular, even dimensional manifolds may carry symplectic structures and odd dimensional manifolds contact structures, the latter the subject of this book by Geiges (Univ. of Cologne, Germany.) Living at the cusp of geometry and topology, these specialized structures have lately generated geometrical techniques offering breakthrough insights into purely topological problems. Symplectic topology has a rich expository literature, but the present volume significantly opens up contact topology to nonspecialists. The first chapter outlines the whole story in a manner attractive for advanced undergraduates (readers will need multilinear algebra and multivariable calculus). The third chapter on relations to knot theory should also particularly interest some undergraduates. Summing Up: Recommended. Upper-division undergraduate through professional collections. D. V. Feldman University of New Hampshire
Copyright American Library Association, used with permission.
Review by Choice Review
