• Smooth Maps of a Foliated Manifold in a Symplectic Manifold

• Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/119/03/0333-0343

• Keywords

Foliations; foliated immersions; foliated symplectic forms.

• Abstract

Let 𝑀 be a smooth manifold with a regular foliation $\mathcal{F}$ and a 2-form 𝜔 which induces closed forms on the leaves of $\mathcal{F}$ in the leaf topology. A smooth map $f:(M,\mathcal{F})\longrightarrow(N, \sigma)$ in a symplectic manifold $(N, \sigma)$ is called a foliated symplectic immersion if 𝑓 restricts to an immersion on each leaf of the foliation and further, the restriction of $f^∗\sigma$ is the same as the restriction of 𝜔 on each leaf of the foliation.

If 𝑓 is a foliated symplectic immersion then the derivative map $Df$ gives rise to a bundle morphism $F:TM\longrightarrow TN$ which restricts to a monomorphism on $T\mathcal{F}\subseteq TM$ and satisfies the condition $F^∗\sigma=\omega$ on $T\mathcal{F}$. A natural question is whether the existence of such a bundle map 𝐹 ensures the existence of a foliated symplectic immersion 𝑓. As we shall see in this paper, the obstruction to the existence of such an 𝑓 is only topological in nature. The result is proved using the ℎ-principle theory of Gromov.

• Author Affiliations

1. Statistics and Mathematics Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata 700 108, India
2. Department of Pure Mathematics, University College of Science, University of Calcutta, 35, P. Barua Sarani, Kolkata 700 019, India

• Proceedings – Mathematical Sciences

Volume 132, 2022
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Continuous Article Publishing mode

• Editorial Note on Continuous Article Publication

Posted on July 25, 2019