Question about the dimension of a Contact (Symplectic) manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:07:32Z http://mathoverflow.net/feeds/question/56127 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56127/question-about-the-dimension-of-a-contact-symplectic-manifold Question about the dimension of a Contact (Symplectic) manifold Phi Le 2011-02-21T01:52:29Z 2011-02-21T09:52:21Z <p>I am reading about contact geometry and I have a question: Why do we only consider contact structure of an odd-dimension manifold? and the same question for definition of symplectic geometry?</p> <p>I think for the contact geometry case, a reason is that we want $\alpha \wedge (d\alpha)^n$ to be a volume form. Am I right? I am not sure about that.</p> <p>Thank you so much for your help.</p> <p>P.S there is no tag for Contact-geometry.</p> http://mathoverflow.net/questions/56127/question-about-the-dimension-of-a-contact-symplectic-manifold/56157#56157 Answer by José Figueroa-O'Farrill for Question about the dimension of a Contact (Symplectic) manifold José Figueroa-O'Farrill 2011-02-21T09:52:21Z 2011-02-21T09:52:21Z <p>And by popular request, here's my comment as an answer :)</p> <p>Your guess is correct. Contact structures are structures associated to a one-form $\alpha$ with maximal rank. There are two cases:</p> <ol> <li><p>for odd rank, you want $\alpha∧(d\alpha)^k$ to be nowhere vanishing for the largest possible $k$ allowed by dimension, and</p></li> <li><p>for even rank, you want the same condition on $(d\alpha)^k$.</p></li> </ol> <p>In the former case you have a contact structure and in the latter an exact symplectic structure.</p> <p>More generally, symplectic forms are nondegenerate by definition. You can understand nondegeneracy of a 2-form $\omega$ pointwise, where it turns into the statement that an antisymmetric matrix has nonzero determinant. This can only happen if the dimension is even.</p> <p>I'm assuming finite-dimensionality throughout. There is a reasonably well-developed theory of infinite-dimensional symplectic manifolds and presumably also of contact manifolds.</p>