Download Geometry in Nature by Vagn Lundsgaard Hansen PDF
By Vagn Lundsgaard Hansen
1. Geometric varieties in Nature. 1. Spirals and the fantastic Snail. 2. The Helix and the Twining Vine. three. The Geometry of cleaning soap motion pictures. four. The Geometry of Tiled Surfaces. five. The common Polyhedra
2. The Topology of Surfaces. 1. a few regular Surfaces. 2. The Projective aircraft and the Klein Bottle. three. what's a Closed floor? four. Orientable and Non-Orientable Surfaces. five. hooked up Sum of Closed Surfaces. 6. category of Closed Surfaces. 7. Higher-Dimensional Manifolds and Poincare's Conjecture
three. The Topology of Catastrophes. 1. The beginning of disaster idea. 2. Singularities: Mappings of the airplane into the airplane. three. The Fold disaster. four. The Cusp disaster. five. Thom's Theorem for platforms with keep an eye on Variables and One nation Variable. 6. a few functions of the Cusp disaster. 7. the maths in the back of the versions of disaster thought. eight. The Seven basic Catastrophes in Space-Time. nine. a few basic comments relating Applications.
four. Geometry and the actual global. 1. On arithmetic and Its Greek Legacy. 2. Greek Astronomy and the Ptolemaic procedure. three. The Copernican international, Tycho Brahe and Kepler. four. The step forward of contemporary traditional technological know-how. five. Newton and Gravitation
five. Geometry and glossy Physics. 1. Maxwell and the Electromagnetic conception. 2. Einstein's thought of Relativity. three. Minkowski Space-Time and the specified conception of Relativity. four. Curvature and Gravitation: the final idea of Relativity. five. The Physics of simple debris. 6. Fiber Bundles and Parallel Displacement in Fiber Bundles. 7. Gauge Theories and String Theories.
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Example text
Given J on the bundle V we extend it by C-linearity to an endomorphism Jc of V'. Since (Jc)2 = -Id, it determines eigenbundles V', V" associated with the eigenvalues ± i; and we have the direct sum decomposition Vc = V' ® V". ) There is a similar decomposition of the dual bundle (V *)c; and these induce decompositions of the various tensor product bundles. In particular, A'(V *)c = p+4=r F, VP. 9 l / with V'- ° = V' and V °' 1 = V ". 3) An Hermitian metric on (ti, J) -+ M is a Riemannian metric a such that
A holomorphic map 0: M -> P1(C) = S2 is just a meromorphic function. The Riemann-Roch theorem and the theory of Weierstrass points give much information concerning their existence. For instance, (1) there is no meromorphic function of Brouwer degree d = 1 on a Riemann surface of genus p > 1; (2) if p = 1 or 2, then such functions exist for all d > 2; (3) if p 3 3, then such functions exist for all d > p+ 1. 38 38 J. EELLS AND L. , there is a meromorphic function of degree 2), then all even degrees are represented, but no odd d < p.
5) below. For n > 3 the energy functional has infimum 0, and hence does not achieve its absolute minimum in any com- ponent of degree k 4 0 [45]. In fact, the pth harmonic suspension constructed above has Morse index > p for p = 5, 6 [206]. 5) with m = 26. We also have harmonic maps of degree ±4 for n = 15, 27, 51; and harmonic maps of all even degrees for n=9. By various such manipulations applied to the examples above we can obtain other harmonic maps between spheres representing nontrivial homotopy classes [212].