From: Sascha.Unzicker@lrz.uni-muenchen.de Newsgroups: sci.math Subject: Re: De Rham Theorem : applications to physics Date: 26 May 1998 14:41:57 GMT mo31g111 writes: >Hi everybody, > I'm doing a memoir on De Rham Theorem. I would like to study some of >its applications to physics. If somebody knows such applications it >would be very nice of him to communicate it. > By advance thanks you very much. Hi, this old posting of mine may be interesting: Cohomology tells you something about the connectedness of a manifold. You have to know the difference between a *closed* and *exact* differential form. A form w is closed, if its exterior derivative vanishes: d w=0. A form w is exact, if its the derivative of another form: d v =w. A exact form is always closed, since dd v=0 is valid for all forms (this contains Div Curl=0, Curl Grad=0, etc). But the contrary is not true, and cohomology measures the 'failure' of the statement closed-> exact. An example : Take the following 1-form w (vector field) in R2\{0}: (-y, x)/(x^2+y^2) w is closed, d w=0 (Curl w=0 in ordinary vector analysis), but its not exact, because there is no global potential v with d v=w! According to a theorem of de Rham, this analytical fact is related to the (topological!) property that R2\{0} is double connected. You can now ask: how meany independent 1-forms (like the above one) (or p-forms)exist, that are closed, but not exact? This number is the so-called 1st (or pth) betti number, a topological invariant that you can calculate from simplicial complexes or CW-complexes. The alternating sum of betti numbers is the Euler characteristic. The cohomolgy groups form with the exterior product a cohomology ring. Literature: 1)Samuel Goldberg, Curvature and Homology (Dover reprint 1962) 2)Bishop & Goldberg, Tensor analysis on manifolds (Dover reprint) 3)M.Nakahara, Geometry, Topology and Physics (Inst. of Phys. publ, 1995) chap. 6. Alexander http://www.lrz-muenchen.de/~u7f01bf/WWW/dg.html