William M Boothby - Collected Quotes

"Apart from its own intrinsic interest, a knowledge of differentiable manifolds has become useful-ven 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)

"Curves and surfaces in Euclidean space were studied since the earliest days of geometry and, after they were invented, both analytic geometry and calculus were systematically used in these studies. However, the discoveries of Gauss, announced in 1827, profoundly altered the course of differential geometry and pointed the way to the concept of abstract difierentiable manifolds-the underlying spaces of every geometry and of other important mathematical theories as well. In his celebrated 'Theorema Egregium' Gauss showed that there is a measure of curvature of a surface (now called the Gaussian curvature) which depends only on one‘s ability to measure the lengths of curves on the surface. This means that this curvature is unchanged by alterations of shape of the surface which leave arclength unchanged. [...] This discovery of an 'inner' geometry, independent of the shape of the surface in E3, led very naturally toward the invention of abstract surfaces (2-manifolds) on which a measure of arclength is (somehow) provided." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"One of the most basic ideas used in the study of mappings from one space to another is that of homotopy. Two mappings are said to be homotopic if one can be 'deformed' into the other through a one-parameter family of mappings between the same spaces. Sometimes further conditions are imposed on the family of mappings [...]" (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)