# Download A C^2 Finite Element and Interpolation by Gao J. PDF

By Gao J.

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Any Grassmannian can be embedded in a unitary group U(n). In Sect. 3, we see how Gauss transforms are a special case of K. Uhlenbeck’s construction [Uh] of adding a uniton, also called ﬂag transform, which provides a method of modifying harmonic maps from a surface to the unitary group to get new ones. This process gives all harmonic 2-spheres in the unitary group as a ‘product’ of unitons. We 1991 Mathematics Subject Classiﬁcation. Primary 53C42; Secondary 58E20. Key words and phrases. Diﬀerential geometry, harmonic map, uniton, Grassmannian model, loop group, nonlinear sigma model.

1. Harmonic maps between Riemannian manifolds. Let (M, g) and (N, h) be smooth Riemannian manifolds and let ϕ : (M, g) → (N, h) be a smooth map between them. The energy or Dirichlet integral of ϕ over a compact domain D of M is deﬁned by E(ϕ, D) = 1 2 D |dϕ|2 ωg EXPLICIT CONSTRUCTIONS OF HARMONIC MAPS 43 3 where ωg = volume measure and, for any p ∈ M , |dϕp |2 is the Hilbert–Schmidt square norm of the diﬀerential dϕp of ϕ at p. In local coordinates, this is given by β |dϕp |2 = g ij hαβ ϕα i ϕj .

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