Zero-energy fields on complex projective space.

*(English)*Zbl 1276.53080In 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.

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.

Reviewer: Paolo Dulio (Milano)