Elasticae in Killing submersions

Classical elastic curves (elastica) are variational objects with many applications in physics and engineering. Elastica in real space forms are well understood, but in other ambient spaces there are few known explicit examples, except geodesics. In this work, we study elastica living in the total space of a Killing submersion focusing on those curves whose osculating plane forms a constant angle with the vertical foliation (slant elastica). First, we compute the Euler–Lagrange equations for elastica and construct new examples of slant elastica in Killing submersions. Then, we completely classify the two main families of slant elastica in Bianchi–Cartan–Vranceanu ambient spaces (giving also explicit parametrizations).