Random Math Notes: Friedrich Hirzebruch, Theorema Egregium, … (2012-06-06)
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Friedrich Ernst Peter Hirzebruch (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as “the most important mathematician in the Germany of the postwar period.”
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a complex manifold in the sense above (which can be specified as an integrable complex manifold), and an almost complex manifold.
Holomorphic = Holomorphic function. Quote:
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. The existence of a complex derivative is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series.
The term analytic function is often used interchangeably with “holomorphic function”, although the word “analytic” is also used in a broader sense to describe any function (real, complex, or of more general type) that is equal to its Taylor series in a neighborhood of each point in its domain. The fact that the class of complex analytic functions coincides with the class of holomorphic functions is a major theorem in complex analysis.
Holomorphic functions are also sometimes referred to as regular functions or as conformal maps. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase “holomorphic at a point z0” means not just differentiable at z0, but differentiable everywhere within some neighborhood of z0 in the complex plane.
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.
Two most beautiful awe-inspiring theorem i learned from Richard Palais are:
Gauss's Theorema Egregium (Latin: “Remarkable Theorem”) is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces. The theorem says that the Gaussian curvature of a surface can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient 3-dimensional Euclidean space. Thus the Gaussian curvature is an intrinsic invariant of a surface.
Started to write the above, didn't finish….