Two Embedding Theorems

Abstract

We first consider pairs $(\mathcal{N},\mathcal{T} )$ where $\mathcal{N}$ is a closed connected smooth manifold and $\mathcal{T}$ a nowhere vanishing smooth real vector field on $\mathcal{N}$ that admits an invariant metric and shows that there is an embedding $F : \mathcal{N} \rightarrow {S}^{2N-1} \subset {\mathbb{C}}^{N}$ for some N mapping $\mathcal{T}$ to a vector field of the form $\mathcal{T}\prime = \mathrm{i}\sum _{j=1}^{N}{\tau }_{j}\big{(}{z}^{j} \frac{\partial } {\partial {z}^{j}} -{\overline{z}}^{j} \frac{\partial } {\partial {\overline{z}}^{j}}\big{)}$ for some τ _j ≠0. We further consider pairs $(\mathcal{N},\mathcal{T} )$ with the additional datum of an involutive subbundle $\overline{\mathcal{V}} \subset \mathbb{C}T\mathcal{N}$ such that $\mathcal{V} + \overline{\mathcal{V}} = \mathbb{C}T\mathcal{N}$ and $\mathcal{V}\cap \overline{\mathcal{V}} ={ \mbox{ span}}_{\mathbb{C}}\mathcal{T}$ for which there is a section β of the dual bundle of $\overline{\mathcal{V}}$ such that $\langle \beta ,\mathcal{T}\rangle = -\mathrm{i}$ and

$$X\langle \beta ,Y \rangle - Y \langle \beta ,X\rangle -\langle \beta ,[X,Y ]\rangle = 0\quad \text{ whenever }X,Y \in {C}^{\infty }(\mathcal{N};\overline{\mathcal{V}}).$$

Then $\overline{\mathcal{K}} =\ker \beta$ is a CR structure, and we give necessary and sufficient conditions for the existence of a CR embedding of $\mathcal{N}$ (with a possibly different, but related, CR structure) into S ^{2N − 1} mapping $\mathcal{T}$ to $\mathcal{T}\prime$. The first result is an analogue of the fact that for any line bundle $L \rightarrow \mathcal{B}$ over a compact base, there is an embedding $f : \mathcal{B}\rightarrow { \mathbb{C}\mathbb{P}}^{N-1}$ such that L is isomorphic to the pullback by f of the tautological line bundle $\Gamma \rightarrow { \mathbb{C}\mathbb{P}}^{N-1}$. The second is an analogue of the statement in complex differential geometry that a holomorphic line bundle over a compact complex manifold is positive if and only if one of its tensor powers is very ample.

Mathematics Subject Classification (2010): Primary 32V30, 57R40. Secondary 32V10, 32V20, 32Q15, 32Q40