**Manifold**- A generalization of -dimensional space in which a neighborhood of each point, called its*chart*, looks like Euclidean space. The charts are related to each other by Cartesian coordinate transformations and comprise an*atlas*for the manifold. The atlas may be non-trivially connected; there are round-trip tours of a manifold that cannot be contracted to a point. The surface of a donut, called a*torus*, is a familiar non-trivial 2D manifold.**Submanifold, ambient space**. A submanifold is a subset of a manifold, its ambient space, for which each point has a chart in which the submanifold looks like a linear subspace of lower dimension. A common knot is a 1-dimensional submanifold of its 3-dimensional ambient space.**Homotopy**. A continuous deformation of a mathematical object which preserves its topological integrity but may develop self-intersections and even worse singularities. There is a homotopy that takes a teapot to a torus (a sphere with a hole). There is another deforming it to a point.**Isotopy**. A homotopy of an object produced by a deformation of the ambient space, so therefore the object cannot develop new self-intersections. The deformation of the teapot to a torus is an isotopy, but the deformation to a point is not.**Embedding.**The parametrization of a submanifold by means of a standard model. A knotted sphere in 4-space is an embedding of the familiar round sphere. Whitney's theorem says that an -dimensional manifold is guaranteed to have an embedding in Euclidean -space.**Immersion.**A locally (but not globally) smoothly invertible mapping of one manifold into another. The image may have self-intersections; the figure-8 is an immersion of the circle in 2D.**Minimal Surface**. A surface that locally has the smallest area given a particular topological shape for it, and possibly, constrained by a fixed boundary (soap-films) or prescribed behavior at infinity.**Steepest Descent Method**. A particular way of guiding an isotopy of an embedded surface to one which minimizes a function that measures its shape. Moving down the gradient of the area function often terminates at a minimal surface.

Thu Sep 21 19:17:33 CDT 1995