Logarithmic connections on principal bundles over a Riemann surface

Let [Formula: see text] be a holomorphic principal [Formula: see text]-bundle on a compact connected Riemann surface [Formula: see text], where [Formula: see text] is a connected reductive complex affine algebraic group. Fix a finite subset [Formula: see text], and for each [Formula: see text] fix [Formula: see text]. Let [Formula: see text] be a maximal torus in the group of all holomorphic automorphisms of [Formula: see text]. We give a necessary and sufficient condition for the existence of a [Formula: see text]-invariant logarithmic connection on [Formula: see text] singular over [Formula: see text] such that the residue over each [Formula: see text] is [Formula: see text]. We also give a necessary and sufficient condition for the existence of a logarithmic connection on [Formula: see text] singular over [Formula: see text] such that the residue over each [Formula: see text] is [Formula: see text], under the assumption that each [Formula: see text] is [Formula: see text]-rigid.

Criterion for connections on principal bundles over a pointed Riemann surface

We investigate connections, and more generally logarithmic connections, on holomorphic principal bundles over a compact connected Riemann surface.

Variations on the Role of Principal Connections in Robotic Locomotion

Diverse problems in robotic locomotion have previously been modeled in terms of connections on principal bundles. Ordinarily, one identifies points in the base manifold of such a bundle with different internal configurations of a robot, identifies points in the fiber over a given base point with different positions and orientations of the robot in its environment, and assumes control to be applied in the former but not the latter. We examine ways in which the ordinary application of this theory may be adapted to problems in aquatic and terrestrial locomotion that fail to accommodate the preceding description.