Conformal Killing forms on totally umbilical submanifolds (Q328614)

For a \(C^{\infty}\)-manifold \(M\) with a pseudo-Riemannian metric \(g\) and Levi-Civita connection \(\nabla\), an \(r\)-form \(\omega\) on \(M\) is called a conformal Killing form if it satisfies the differential equation: \[ \nabla\omega-\frac{1}{r+1}d\omega+g\wedge\theta=0 \] for some \((r-1)\)-form \(\theta\). It is known that on a Riemannian manifold \(M\) of nonzero constant curvature, the vector space of conformal Killing \(r\)-forms \(\mathbf T^r(M,\mathbb R)\) can be decomposed as \[ \mathbf T^r(M,\mathbb R)=\mathbb K^r(M,\mathbb R)\oplus\mathbb P^r(M,\mathbb R), \] where \(\mathbf K^r(M,\mathbb R)\) is the vector space of Killing \(r\)-forms and \(\mathbf P^r(M,\mathbb R)\) is the vector space of closed Killing \(r\)-forms. In this paper, the authors study the global existence of conformal Killing forms on compact and orientable submanifolds of pseudo-Riemannian manifolds, and determine exact upper bounds of the dimension of the above vector spaces.