On Manifolds (1975-1999)

"Apart from its own intrinsic interest, a knowledge of differentiable manifolds has become useful - even mandatory - in an ever-increasing number of areas of mathematics and of its applications. This is not too surprising, since differentiable manifolds are the underlying, if unacknowledged, objects of study in much of advanced calculus and analysis. Indeed, such topics as line and surface integrals, divergence and curl of vector fields, and Stokes's and Green's theorems find their most natural setting in manifold theory. But however natural the leap from calculus on domains of Euclidean space to calculus on manifolds may be to those who have made it, it is not at all easy for most students." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"Our interest is in pointing out that new surfaces, that is, 2-manifolds, can be formed by fastening together manifolds with boundary along their boundaries, that is, by identifying points of various boundary components by a homeomorphism, assuming of course the necessary condition that such components are homeomorphic." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"There is, however, a unique feature of the tangent spaces of Euclidean space which is not shared by the tangent spaces at points of manifolds; the tangent spaces at any two points of Euclidean space are naturally isomorphic, that is, there is an isomorphism determined in some unique fashion by the geometry of the space - not chosen by us." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"A manifold M of dimension n, or n-manifold, is a topological space with the following properties: (i) M is Hausdorff, (ii) M is locally Euclidean of dimension n, and (iii) M has a countable basis of open sets." (William M Boothby, "An introduction to differentiable manifolds and Riemannian geometry" 2nd Ed., 1986)

"An attractor that consists of an infinite number of curves, surfaces, or higher-dimensional manifolds - generalizations of surfaces to multidimensional space - often occurring in parallel sets, with a gap between any two members of the set, is called a strange attractor." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Determination of transition functions makes it possible to restore the whole manifold if individual charts and coordinate maps are already given. Glueing functions may belong to different functional classes, which makes it possible to specify within a certain class of topological manifolds more narrow classes of smooth, analytic, etc. manifolds." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Roughly speaking, manifolds are geometrical objects obtained by glueing open discs" (balls) like a papier-mache is glued of small paper scraps. To this end, one first prepares a clay or plastecine figure which is then covered with several sheets of paper scraps glued onto one another. After the plasticine is removed, there remains a two-dimensional surface." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"[...] a manifold is a set M on which 'nearness' is introduced (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"It is commonly said that the study of manifolds is, in general, the study of the generalization of the concept of surfaces. To some extent, this is true. However, defining it that way can lead to overshadowing by 'figures' such as geometrical surfaces." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Algebraic topology studies properties of a narrower class of spaces, - basically the classical objects of mathematics: spaces given by systems of algebraic and functional equations, surfaces lying in Euclidean space, and other sets which in mathematics are called manifolds. Examining the narrower class of spaces permits deeper penetration into their structure. (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)