Contact Forms
A differential 1-form θ on the space Jr(π) is called a contact form (i.e. ) if it is pulled back to the zero form on M by all prolongations. In other words, if, then if and only if, for every open submanifold W ⊂ M and every σ in ΓM(π)
The distribution on Jr(π) generated by the contact forms is called the Cartan distribution. It is the main geometrical structure on jet spaces and plays an important role in the geometric theory of partial differential equations. The Cartan distributions are not involutive and are of growing dimension when passing to higher order jet spaces. Surprisingly though, when passing to the space of infinite order jets J∞ this distribution is involutive and finite dimensional. Its dimension coinciding with the dimension of the base manifold M.
Read more about this topic: Jet Bundle
Famous quotes containing the words contact and/or forms:
“I stand in awe of my body, this matter to which I am bound has become so strange to me. I fear not spirits, ghosts, of which I am one,—that my body might,—but I fear bodies, I tremble to meet them. What is this Titan that has possession of me? Talk of mysteries! Think of our life in nature,—daily to be shown matter, to come in contact with it,—rocks, trees, wind on our cheeks! the solid earth! the actual world! the common sense! Contact! Contact! Who are we? where are we?”
—Henry David Thoreau (1817–1862)
“I am not so foolish as to declaim against forms. Forms are as essential as bodies; but to exalt particular forms, to adhere to one form a moment after it is outgrown, is unreasonable, and it is alien to the spirit of Christ.”
—Ralph Waldo Emerson (1803–1882)