By Alexei Kushner
Tools from touch and symplectic geometry can be utilized to unravel hugely non-trivial nonlinear partial and usual differential equations with no resorting to approximate numerical equipment or algebraic computing software program. This booklet explains how it truly is performed. It combines the readability and accessibility of a complicated textbook with the completeness of an encyclopedia. the elemental principles that Lie and Cartan built on the finish of the 19th century to rework fixing a differential equation right into a challenge in geometry or algebra are the following remodeled in a unique and glossy manner. Differential equations are regarded as part of touch and symplectic geometry, in order that the entire equipment of Hodge-deRham calculus could be utilized. during this approach a large type of equations should be tackled, together with quasi-linear equations and Monge-Ampere equations (which play a huge function in sleek theoretical physics and meteorology).
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Sample text
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].