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Zero-energy fields on complex projective space. (English) Zbl 1276.53080
In this paper, the authors consider a complex projective space $$\mathbb{C}\mathbb{P}_n$$ with its Fubini-Study metric.
If $$\omega_{ab...c}$$ is a smooth symmetric covariant tensor field defined on a Riemannian manifold $$M$$, $$\gamma$$ is a smooth oriented curve on $$M$$, and $$X^a$$ is the unit vector field defined along $$\gamma$$ and tangent to $$\gamma$$ consistent with its orientation, then, by integrating $$X^aX^b...X^c\omega_{ab...c}$$ along $$\gamma$$ with respect to arc length, a real number $$\int_{\gamma}\omega_{ab...c}$$ is obtained.
On $$\mathbb{C}\mathbb{P}_n$$, the $$X$$-ray transform on symmetric tensor fields associates to $$\omega_{ab...c}$$ the function $$\gamma\mapsto\int_{\gamma}\omega_{ab...c}$$, defined on the space of geodesics of $$\mathbb{CP}_n$$. The kernel of the $$X$$-ray transform consists of all fields such that $$\int_{\gamma}\omega_{ab...c}=0$$.
It is easy to see that fields of the form $$\nabla_{(a\Phi_{b...c})}$$ belong to the kernel, where $$\Phi_{b...c}$$ is a symmetric covariant tensor field, $$\nabla_a$$ is the Levi-Civita connection, and round brackets denote symmetrization over the enclosed indices.
The main result of the paper shows that the converse is also true, namely, that a smooth symmetric covariant tensor field $$\omega_{ab...c}$$ of valence $$p\geq 1$$ belonging to the kernel of the $$X$$-ray transform must be of the form $$\nabla_{(a\Phi_{b...c})}$$ for some smooth symmetric field $$\Phi_{b...c}$$ of valence $$p-1$$.
Such a result is derived from the corresponding known statement for real projective space $$\mathbb{R}\mathbb{P}_n$$ with its usual round metric.
The approach bases on the fact that the standard embedding of $$\mathbb{R}\mathbb{P}_n$$ in $$\mathbb{C}\mathbb{P}_n$$ induced by the embedding of $$\mathbb{R}^{n+1}$$ in $$\mathbb{C}^{n+1}$$ is totally geodesic, and the same holds also for all its translates under the group of isometries.

##### MSC:
 53C65 Integral geometry 44A12 Radon transform
##### Keywords:
integral geometry; $$X$$-ray transform
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