basicelements of differential geometry and topology

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The research of third-dimensional areas brings jointly components from numerous parts of arithmetic. the main striking are topology and geometry, yet parts of quantity concept and research additionally make appearances. long ago 30 years, there were outstanding advancements within the arithmetic of three-dimensional manifolds.

Extra info for basicelements of differential geometry and topology

Example text

3 (Hamiltonian vector field). Suppose (M, ω) is a symplectic manifold, and suppose H be a function H : M → R. Then the Hamiltonian vector field induced by H is the unique (see next paragraph) vector field XH ∈ X (M ) determined by the condition dH = ι XH ω. In the above, ι is the contraction mapping ι X : Ωr M → Ωr−1 M defined by (ιX ω)(·) = ω(X, ·). To see that XH is well defined, let us consider the mapping X → ω(X, ·). By non-degeneracy, it is injective, and by the rank-nullity theorem, it is surjective, so the Hamiltonian vector field X H is uniquely determined.

Then for all x ∈ U , t ∈ I, we have (Φ∗t ω)x = ωx , where Φt = Φ(t, ·). Proof. As Φ0 = idM , we know that the relation holds, when t = 0. Therefore, let us fix x ∈ U , a, b ∈ Tx M , and consider the function r(t) = (Φ∗t ω)x (a, b) with t ∈ I. Then r (t) = = = d r(s + t) ds s=0 d ∗ (Φs ω)y (a , b ) ds LXH ω y (a , b ), s=0 where y = Φt (x), a = (DΦt )(a), b = (DΦt )(b), and the last line is the definition of the Lie derivative. Using Cartan’s formula, L X = ιX ◦ d + d ◦ ιX , we have LXH ω = ιXH dω + dιXH ω = ιXH 0 + ddH = 0, so r (t) = 0, and r(t) = r(0) = ωx (a, b).

The cotangent bundle T ∗ M \ {0} of manifold M is a symplectic manifold with a symplectic form ω given by ω = dθ = dxi ∧ dξi , where θ is the Poincar´e 1-form θ ∈ Ω1 T ∗ M \ {0} . ∂ i ∂ i ∂ Proof. It is clear that ω is closed. If X = a i ∂x i + b ∂ξ , and Y = v ∂xi + i wi ∂ξ∂ i , then ω(X, Y ) = a · w − b · v. By setting v = w we obtain a = b, and by setting w = 0 we obtain a = b = 0. The next example shows that we can always formulate Hamilton’s equations on the cotangent bundle. This motivates the name for X H .