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contact manifold (Definition)

Let $ M$ be a smooth manifold and $ \alpha$ a one form on $ M$. Then $ \alpha$ is a contact form on $ M$ if

  1. for each point $ m \in M$, $ \alpha_{m} \neq 0$ and
  2. the restriction $ d\alpha_{m}\vert _{\ker{\alpha_{m}}}$ of the differential of $ \alpha$ is nondegenerate.

Condition 1 ensures that $ \xi=\ker{\alpha}$ is a subbundle of the vector bundle $ TM$. Condition 2 equivalently says $ d\alpha$ is a symplectic structure on the vector bundle $ \xi \to M$. A contact structure $ \xi$ on a manifold $ M$ is a subbundle of $ TM$ so that for each $ m \in M$, there is a contact form $ \alpha$ defined on some neighborhood of $ m$ so that $ \xi = \ker{\alpha}$. A co-oriented contact structure is a subbundle of $ TM$ of the form $ \xi=\ker{\alpha}$ for some globally defined contact form $ \alpha$.

A (co-oriented) contact manifold is a pair $ (M,\xi)$ where $ M$ is a manifold and $ \xi$ is a (co-oriented) contact structure. Note, symplectic linear algebra implies that $ \dim{M}$ is odd. If $ \dim{M} = 2n+1$ for some positive integer $ n$, then a one form $ \alpha$ is a contact form if and only if $ \alpha \wedge (d\alpha)^{n}$ is everywhere nonzero.

Suppose now that $ (M_1,\xi_1=\ker{\alpha_1})$ and $ (M_2,\xi_2=\ker{\alpha_2})$ are co-oriented contact manifolds. A diffeomorphism $ \phi:M_1 \to M_2$ is called a contactomorphism if the pullback along $ \phi$ of $ \alpha_2$ differs from $ \alpha_1$ by some positive smooth function $ f:M_1 \to \mathbb{R}$, that is, $ \phi^{*}\alpha_{2} = f\alpha_{1}$.

Examples:

  1. $ \mathbb{R}^3$ is a contact manifold with the contact structure induced by the one form $ \alpha = dz + xdy$.
  2. Denote by $ \mathbb{T}^{2}$ the two-torus $ \mathbb{T}^{2} = S^1 \times S^1$. Then, $ \mathbb{R} \times \mathbb{T}^2$ (with coordinates $ t,\theta_1,\theta_2$) is a contact manifold with the contact structure induced by $ \alpha = \cos{t\theta_1}+\sin{t\theta_2}$.



"contact manifold" is owned by RevBobo.
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See Also: symplectic manifold

Also defines:  contact structure, contact form, contactomorphism
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Cross-references: coordinates, induced, smooth function, pullback, diffeomorphism, integer, positive, odd, implies, linear algebra, neighborhood, structure, vector bundle, subbundle, nondegenerate, restriction, point, smooth manifold
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This is version 1 of contact manifold, born on 2003-06-27.
Object id is 4409, canonical name is ContactManifold.
Accessed 5348 times total.

Classification:
AMS MSC53D10 (Differential geometry :: Symplectic geometry, contact geometry :: Contact manifolds, general)

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