By Jürgen Jost

ISBN-10: 0387133399

ISBN-13: 9780387133393

Show description

Read or Download Harmonic Maps Between Surfaces (with a Special Chapter on Conformal Mappings) PDF

Best differential geometry books

Download e-book for iPad: The topology of fibre bundles by Norman Steenrod

Fibre bundles, now a vital part of differential geometry, also are of serious value in sleek physics - corresponding to in gauge thought. This booklet, a succinct advent to the topic via renown mathematician Norman Steenrod, was once the 1st to provide the topic systematically. It starts off with a common creation to bundles, together with such themes as differentiable manifolds and masking areas.

I. Chavel, H.M. Farkas's Differential geometry and complex analysis: a volume PDF

Chavel I. , Farkas H. M. (eds. ) Differential geometry and intricate research (Springer, 1985)(ISBN 354013543X)(236s)

Download e-book for iPad: Theorems on regularity and singularity of energy minimizing by Leon Simon

The purpose of those lecture notes is to offer an basically self-contained advent to the elemental regularity concept for power minimizing maps, together with fresh advancements in regards to the constitution of the singular set and asymptotics on method of the singular set. really expert wisdom in partial differential equations or the geometric calculus of adaptations is no longer required.

Additional resources for Harmonic Maps Between Surfaces (with a Special Chapter on Conformal Mappings)

Sample text

We sum up our discussion with the basic definition. Definition 7. A noncommutative spin geometry is a real spectral triple (A, H, D; , J ) or (A, H, D; J ), according as its dimension is even or odd, that satisfies the seven axioms set out above. Riemannian spin manifolds provide the commutative examples. It is not hard to manufacture noncommutative examples with finite-dimensional matrix algebras [136], [165]; these are zero-dimensional geometries in the sense of Axiom 1. In the next chapter we study a more elaborate noncommutative example which, like the Riemann sphere, has dimension two.

Also, the coefficient of logarithmic divergence is / −2 ) σN (D = 2. N →∞ log N / −2 = lim −D As we shall see later on, this coefficient is 1/2π times the area for any 2-dimensional surface, so the area of the sphere is hereby computed to be 4π . 3 The first-order condition Axiom 2 (Order one). For all a, b ∈ A, the following commutation relation holds: [[D, a], J b∗ J † ] = 0. 4) This could be rewritten as [[D, a], b ] = 0 or as [[D, π(a)], π (b)] = 0. 2) and the Jacobi identity, we see that this condition is symmetric in the representations π and π , since [a, [D, b ]] = [[a, D], b ] + [D, [a, b ]] = −[[D, a], b ] = 0.

Now the Eisenstein series G2k (τ ) := m,n 1 , (m + nτ )2k with primed summation ranging over integer pairs (m, n) = (0, 0), converges absolutely for k > 1 and only conditionally for k = 1. We shall see below that −2 in fact diverges logarithmically, thereby establishing the two-dimenm,n |m+nτ | sionality of the geometry. The orientation cycle. In terms of the generators u = e2π iφ1 , v = e2π iφ2 of A0 , the volume form on the torus T2 is dφ1 ∧ dφ2 = (2π i)−2 u−1 v −1 du ∧ dv, with the corresponding Hochschild cycle: (2π i)−2 v −1 u−1 ⊗ u ⊗ v − u−1 v −1 ⊗ v ⊗ u .

Download PDF sample

Harmonic Maps Between Surfaces (with a Special Chapter on Conformal Mappings) by Jürgen Jost


by William
4.3

Rated 4.73 of 5 – based on 23 votes