Sentence examples for a flat connection from inspiring English sources
The construction of the geodesic loop associated with a flat connection is due to Kikkawa [24] and Sabinin [49].
It can be shown that the necessary condition for an infinitesimal loop (which is also sufficient in the case of analytic loops) to be a geodesic loop of a flat connection is that it should satisfy the right alternative property.
Indeed, there are two constructions of the tangent structure to a local (infinitesimal, formal) loop and both are based on the fact that there is a flat connection associated with a local loop, see the next section.
On the tangent bundle ({T} G) there exists a canonical flat connection (nabla ).
In the case of the lower reservoir there is a nearby access road, so only a short and flat connection to it is necessary.
For example, it is possible to produce a one-sided flat connection, which is not producible with any other joining technology.
It can be seen that each flat connection on a neighbourhood of a point comes from a local loop, called the geodesic loop of the connection.
From this construction it is clear that given a flat Sabinin algebra (mathfrak {l}) one can reconstruct, locally, the corresponding flat connection, and, hence, the infinitesimal loop, provided that the operations of (mathfrak {l}) define converging power series.
It would be fair to say that most of the constructions and results on Sabinin algebras mentioned in the previous section were the product of a quest for a Lie theory of flat connections.
Moreover, under an additional geometrical hypothesis, we show that the transformationΦSestablishes an equivalence of categories between coherent DX-modules, modulo flat connections, and coherent DY-modules with regular singularities along an involutive manifoldV, modulo flat connections (hereVis determined by the geometry of the correspondence).
This induces a symplectic structure for the moduli space of flat connections with boundary holonomies lying in prescribed conjugacy classes.
