By Sigurdur Helgason
This writer supplies the 1st systematic exposition of geometric research on Riemannian symmetric areas and its courting to the illustration thought of semisimple Lie teams. The e-book begins with smooth necessary geometry for double fibrations and treats numerous examples intimately. After discussing the idea of Radon transforms and Fourier transforms on symmetric areas, Helgason examines purposes to invariant differential equations on symmetric areas, really power concept and wave equations. The booklet concludes with a bankruptcy on eigenspace representations--that is, representations on answer areas of invariant differential equations. recognized for his high quality expositions, Helgason got the 1988 AMS Steele Prize for his past books Differential Geometry, Lie teams and Symmetric areas and teams and Geometric research. Containing routines (with recommendations) and references to additional effects, this publication will be appropriate for complex graduate classes in glossy crucial geometry, research on Lie teams, or illustration thought of semisimple Lie teams.
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Extra resources for Geometric Analysis on Symmetric Spaces
It is precisely this nice feature which will be altered in the example of the next section. 3 A Dispersive Wave Equation Imagine that you and your friends stood out in a ﬁeld and positioned yourselves so that you spelled out a word when viewed from above. If you all moved in the same direction at the same speed, then to an observer in a helicopter, this word would appear to “travel” across the ﬁeld. On the other hand, if you each moved at a diﬀerent speed, then the word would only be visible brieﬂy and would quickly degenerate into a “mess” to the observer.
7) and I want to produce a solution of the (equivalent) equation 2Ut + 9U Ux = 0 by choosing nonzero constants λ and γ so that U (x, t) = u(λx, γt) is a solution of this new equation. What choices of λ and γ will be sure to work? 7. Perhaps we are taking too simple a deﬁnition of what diﬀerentiates a linear diﬀerential equation from a nonlinear diﬀerential equation in this book. The purpose of this question is merely to emphasize that there are subtler points one might want to consider if this dichotomy is to be taken seriously.
Examples of such equations include ut = uux and ut = u2 + uxxx . In this section, we will learn a simple method for determining the approximate dynamics of a solution6 to these equations over a very short time interval starting from any given initial proﬁle u(x, 0) = u0 (x). If we think of the diﬀerential equation as a rule for determining how things can evolve in time, this gives us a way to see what would happen at later times if we know the shape of the wave at time t = 0 without having to be able to ﬁnd a formula for such a solution.
Geometric Analysis on Symmetric Spaces by Sigurdur Helgason
Categories: Algebraic Geometry