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What is an overtwisted contact structure?

A contact structure ~ on a connected 3-manifold M is called overtwisted if there is a contact embedding of the standard overtwisted disc (A, (1) into (M, 4). As follows from 1.2, an overtwisted structure contains an embedded 2-disc ~ such that the induced foliation 9r has the form as in Fig.

What is differential geometry used for?

Differential geometry is the tool we use to understand how to adapt concepts such as the distance between two points, the angle between two crossing curves, or curvature of a plane curve, to a surface.

What is the standard contact structure in R 3?

The standard contact structure on R3. Each point in R3 has a plane associated to it by the contact structure, in this case as the kernel of the one-form dz ? y dx. These planes appear to twist along the y-axis.

What is contact topology?

Contact and Symplectic Topology Symplectic and contact structures first arose in the study of classical mechanical systems, allowing one to describe the time evolution of both simple and complex systems such as springs, planetary motion, and wave propagation.

What is a contact transformation?

A vector field X on a contact manifold is called a contact (or strict contact) infinitesimal transformation if LX?=??? (or LX?=0), where LX is the Lie derivative along X. Contact transformations are sometimes called tangency transformations.

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