By Sze-Tsen Hu
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Extra info for Homology theory: A first course in algebraic topology
2. 2 The Dirichlet integral, and the set of harmonic maps on an open subset of a Riemannian surface, depend only on the conformal structure. 3, we can always suppose that we have locally conformal coordinates (x, y) ∈ R2 on (M, g). 13) becomes ∂ 2 ui ∂ 2 ui + + Γijk (u) 2 ∂x ∂y 2 ∂uj ∂uk ∂uj ∂uk + ∂x ∂x ∂y ∂y = 0. 13), but an alternative one where we think of N as a submanifold of a Euclidean space. In fact, thanks to the Nash–Moser theorem (, , ), we know that, provided h is C 3 , it is always possible to isometrically embed (N , h) into a vector space RN , with the Euclidean scalar product .
In the case where m = 2, this method no longer works, but Luc Lemaire proved a similar result which is valid for every simply connected open set . 44) which we used in Case A to deduce a conservation law in the case where there exists a Killing vector field. Writing Sβα = g αγ Sγβ , this identity states that, for any vector field X, 2Sαβ ∇ ∂ ∂xβ X α = −(LX g αβ )Sαβ . 37) implies that ∇ ∂α Sβα = 0 for ∂x all β, and thus ∇ ∂ ∂xβ (Sαβ X α ) = Sαβ ∇ ∂ ∂xβ Xα + ∇ 1 = − (LX g αβ )Sαβ . 52) Geometric and analytic setting 28 We see that for the vector field Y = Sαβ X α ∂x∂ β to be covariant divergencefree, it is necessary and sufficient that (LX g αβ )Sαβ = 0 .
1 Let X be a Lipschitz tangent vector field on N , which is an infinitesimal symmetry for L. 29) or equivalently, using the coordinates (x1 , . . , xm ) on Ω such that dµ = ρ(x)dx1 . . dxm , m ∂ α ∂x α=1 ρ(x)X i (u) ∂L (x, u, du) ∂Aiα = 0. 2 (i) In the case where L is the Lagrangian of the harmonic map, this result was first obtained in . (ii) The vector field J defined over Ω by J α = ρ(x) ∂L (x, u, du)X i (u) ∂Aiα is often called the Noether current by physicists. 30) are called conservation laws.
Homology theory: A first course in algebraic topology by Sze-Tsen Hu