Showing papers on "Geometry and topology published in 1978"

A course in differential geometry

Abstract: 0 Calculus in Euclidean Space.- 1 Curves.- 2 Plane Curves: Global Theory.- 3 Surfaces: Local Theory.- 4 Intrinsic Geometry of Surfaces: Local Theory.- 5 Two-dimensional Riemannian Geometry.- 6 The Global Geometry of Surfaces.- References.- Index of Symbols.

A fixed point theorem in symplectic geometry

Abstract: There are a number of fixed point theorems pecuhar to symplectic geometry. A particularly simple example is the theorem tha t any area-preserving mapping yJ of the two-dimensional sphere into itself possesses at least two distinct fixed points (see [6, 8]) although an arbi trary orientation-preserving mapping may have only one single fixed point. In higher dimensions such global theorems are not available, but it is known (see [11]) tha t any symplectic map yJ which is Cl-close to the identi ty map of a simply connected, compact symplectic manifold into itself has at least two fixed points. These fixed points are found as critical points of appropriate functions on the manifold. In t h i s note we will derive a generalization of such a perturbation theorem which has various applications in mechanics. To formulate our result we need some concepts of symplectic geometry: A smooth manifold Z is called symplectic if there exists a non-degenerate closed 2-form ~o on Z; the symplectic manifold consists in fact of the pair (Z, co). I f eo is even exact and given by o~ = d~, a being a 1-form we call (Z, cr an exact symplcctic manifold. The most familiar example of an exact symplectic manifold is the cotangent bundle of any manifold with its natural 1-form. A differentiable mapping ~ of Z into itself is called symplectic if it preserves the two-form co, i.e. if ~0*eo =~o. Similarly, we call a mapping yJ exact symplectic if (Z, ~) is exact and ~f*~-~ is exact, i . e . = d F where F is a function of Z. We apply the same terminology for mappings ~ of an open set D I ~ Z into another D 2 c Z. Of course, every exact symplectic mapping is also symplectic since ~ o * a = ~ + d F implies

On the differential geometry of frame bundles of Riemannian manifolds.

Abstract: Let M be an «-dimensional differentiable manifold of class C°°, TM its tangent bündle and FM its frame bündle. The differential geometry of TM has been studied by many authors and a survey of their results can be found in Yano and Ishihara [13]. On the other band, the differential geometry of FM was first studied by T. Okubo [6], [7]. Recently, the author [4] has considered the complete lift of tensor fields and connections on M toFM.

Algebraic and geometric topology

Abstract: Contains sections on Structure of topological manifolds, Low dimensional manifolds, Geometry of differential manifolds and algebraic varieties, $H$-spaces, loop spaces and $CW$ complexes, Problems.