Varieties of minimal rational tangents of unbendable rational curves subordinate to contact structures
نویسندگان
چکیده
A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus q}$ for some nonnegative integers $p$ and $q$ called an unbendable on $X$. Associated with it the variety of minimal tangents (VMRT) at point $x \in C$, which germ submanifolds $\mathcal{C}^C_{x} \subset \mathbb{P} T_{x} X$ consisting tangent directions small deformations fixing $x$. Assuming that there exists distribution $D TX$ such all are $D$, one asks what kind projective space can be realized as VMRT $\mathcal{C}^{C}_{x} D_{x}$. When contact distribution, well-known necessary condition $\mathcal{C}_{x}^{C}$ should Legendrian respect induced structure $\mathbb{P} We prove this also sufficient condition: we construct $C $D$ D_{x}$ C$ projectively arbitrarily given submanifold. Our construction uses geometry lines Heisenberg group technical ingredient symplectic distributions study has originated from geometric control theory.
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ژورنال
عنوان ژورنال: Journal of The Mathematical Society of Japan
سال: 2022
ISSN: ['1881-1167', '0025-5645']
DOI: https://doi.org/10.2969/jmsj/85868586