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We shall discuss this idea later in the context of harmonic map theory, see chapter 4. 3 Euler–Lagrange equations on Riemannian manifolds 39 Let (M, g) be a Riemannian manifold and φ ∈ F(M). 9) ∂xj ∂xj where ∇ is the Levi-Civita connection on M. Consider a map : M → N, where M and N are Riemannian manifolds. The first is the space of parameters and the second the space of coordinates. , yn ) are coordinates around (p) ∈ N, we define the vector field ; i ∈ X (M) by ;i = ∗ ∂ = ∂xi j ;i ∂ . 10) In the particular case when M = Rt , we obtain the tangent vector field along ˙ (t) = ∗ d .

Such an immersion is called isometric. 19) i=1 where Y = Y i ei , X = Xi ei , and e1 = (1, 0, . . , 0), . . , en = (0, . . , 0, 1). If ∇ is the Levi-Civita connection on M, the second fundamental form of the immersion f is the two-covariant, symmetric tensor field on M h(X, Y ) = ∇˜ X Y − ∇X Y, ∀X, Y ∈ X (M). 20) is called Gauss’s formula, and we have h(X, Y ) = nor (∇˜ X Y ), ∇X Y = tan (∇˜ X Y ), where nor (tan) represents the normal (tangential) component with respect to M. 8 The mean curvature vector field of the submanifold M of Rn is H = 1 T raceg h.

2) where c = min V . Under certain conditions M is supposed to be a connected submanifold of E, so that the scalar product : E × E → R induces a Riemannian metric g on M. 3) 2 which will be the Lagrangian for the harmonic maps and will be considered later. In geometry the notion was introduced by J. H. Sampson, see [13].

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