Show description

To do this, we seek an expression for its exterior derivative, and to understand what such an expression should look like, we proceed as follows. Consider a local trivialization B ∼ = M × G, induced by a choice of section η of B → M whose image is identified with M × {e} ⊂ M × G. The section η is in particular an Rn -valued 1-form on M , and the tautological 1-form is ω = g−1 η ∈ Ω1(B) ⊗ Rn. The exterior derivative of this equation is dω = −g−1 dg ∧ ω + g−1 dη. 7) Note that the last term in this equation is semibasic for B → M , and that the matrix 1-form g −1 dg takes values in the Lie algebra g of G.

This implies that v induces a vector field downstairs on M , whose flow is easily seen to be ϕt . The fact that Lv θ = 0 confirms that ϕt is a contact transformation. We can now examine the effect of ϕt on the invariant Euler-Lagrange systems corresponding to linear Weingarten equations by introducing Ψ2 = π 1 ∧ π2 , Ψ1 = π1 ∧ ω 2 − π2 ∧ ω 1 , Ψ0 = ω 1 ∧ ω 2 . 4. HYPERSURFACES IN EUCLIDEAN SPACE 27 Restricted to a transverse Legendre submanifold over a surface N ⊂ E3 , these give K dA, H dA, and the area form dA of N , respectively.

THE EQUIVALENCE PROBLEM FOR n = 2 41 An example of a hyperbolic Monge-Ampere system, to be studied in more detail in Chapter 4, is the linear Weingarten system for surfaces in E3 with Gauss curvature K = −1. To begin, assume that (M 5 , E) is a hyperbolic Monge-Ampere system. 3) and also dη0 ≡ η1 ∧ η2 + η3 ∧ η4 (mod {I}). 4) According to the following proposition, a hyperbolic Monge-Ampere system is equivalent to a certain type of G-structure, and it is the latter to which the equivalence method directly applies.