On naturality of some construction of connections

Let \(F\) be a bundle functor on the category of all fibred manifolds and fibred maps. Let \(\Gamma\) be a general connection in a fibred manifold \(\mathrm{pr}:Y\to M\) and \(\nabla\) be a classical linear connection on \(M\). We prove that the well-known general connection \(\mathcal{F}(\Gamma,\nabla)\) in \(FY\to M\) is canonical with respect to fibred maps and with respect to natural transformations of bundle functors.

On canonical constructions on connections

We study how a projectable general connection \(\Gamma\) in a 2-fibred manifold \(Y^2\to Y^1\to Y^0\) and a general vertical connection \(\Theta\) in \(Y^2\to Y^1\to Y^0\) induce a general connection \(A(\Gamma,\Theta)\) in \(Y^2\to Y^1\).

The induced connections on total spaces of fibred manifolds

Let Y ? M be a fibred manifold with m-dimensional base and n-dimensional fibres. If m ? 2 and n ? 3, we classify all linear connections A(?, ?, ?) : TY ? J1(TY ? Y) in TY ? Y (i.e., classical linear connections on Y) depending canonically on a system (?, ?, ?) consisting of a general connection ? : Y ? J1Y in Y ? M, a torsion free classical linear connection ? : TM ? J1(TM ? M) on M and a linear connection ? : V Y ? J1(VY ? Y ) in the vertical bundle VY ? Y.

HERMITIAN VECTOR FIELDS AND SPECIAL PHASE FUNCTIONS

We start by analyzing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.