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Unformatted text preview: 5.3. FROBENIUS THEOREM 119 5.3 Frobenius Theorem 5.3.1 Distributions • Definition 5.3.1 Let M be a n-dimensional manifold. A k-dimensional distribution (or a tangent subbundle ) Δ : M → Δ x ⊂ T x M is a smooth assignment to each point x ∈ M a k dimensional subspace Δ x of the tangent space T x M. An submanifold V of M that is everywhere tangent to the distribution is called an integral manifold of the distribution. A k-didmensional distribution Δ is called integrable if at each point x ∈ M there is a k-dimensional integral submanifold of Δ . In other words, the distribution Δ is integrable if everywhere in M there exist local coordinates ( x 1 , . . . , x k , y 1 , . . . , y n- k ) such that the coordinate surfaces y a = c a , a = 1 , . . . , n- k, c a being some constants, are integral manifolds of the distribution Δ . Such a coordinate system is called a Frobenius chart ....
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- Spring '10
- Geometry, Manifold, Differential topology, Frobenius theorem