# NOTE ON NORMAL EMBEDDING

• Published : 2002.05.01
• 30 14

#### Abstract

It was shown by L. Polterovich ([3]) that if L is a totally real submanifold of a symplectic manifold $(M,\omega)$ and L is parallelizable then L is normal. So we try to find an answer to the question of whether there is a compatible almost complex structure J on the symplectic vector bundle $TM$\mid$_{L}$ such that $TL{\cap}JTL=0$ assuming L is normal and parallelizable. Although we could not reach an answer, we observed that the claim holds at the vector space level. And related to the question, we showed that for a symplectic vector bundle $(M,\omega)$ of rank 2n and $E=E_1{\bigoplus}E_2$, where $E=E_1,E_2$are Lagrangian subbundles of E, there is an almost complex structure J on E compatible with ${\omega}$ and $JE_1=E_2$. And finally we provide a necessary and sufficient condition for a given embedding into a symplectic manifold to be normal.

#### Keywords

normal embedding;totally real embedding;submanifold

#### References

1. D. McDuff and D. Salamon, Introduction to symplectic topology, Oxford University Press, 1995.
2. J. Milnor and J. D. Stasheff, Characteristic Classes, Annals of Mathematics Studies,No. 76, Princeton University Press, 1974.
3. L. Polterovich, An obstacle to non-Lagrangian intersections, The Floer memorial volume, Progr. Math., 133, Birkhiiser Basel, 1995, pp. 575-586.
4. J.-C. Sikorav, Quelques proprietes des plongements Lagrangiens, Mem. Soc. Math.Fr., Nouv. Ser. 46 (1991),151-167.