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Micro Squares, Connections and the Lie Bracket of Vector Fields

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Using the exponential laws in the cartesian closed category of microlinear spaces we obtain that the iterated tangent bundle is the bundle of micro squares. There are various ways to structure it as a bundle over the tangent bundle. One way is to see it as a bundle over ordered pairs of tangents by mapping a micro square to its pair of principal axes. This bundle turns out to be an affine bundle over the tangent bundle and plays an important (unifying) role for connections and Lie brackets of vector fields.

Geometrically, a (linear) connection on a tangent bundle is a structure that ‘connects’ infinitesimally neighbouring tangent spaces (respecting their R-linear structures). Any section of the affine bundle of micro squares is considered a connection, since passing from a pair of tangent vectors to the respective micro square amounts to thickening the pair of tangent vectors to an infinitesimal grid, allowing infinitesimal ‘parallel’ transport of one tangent along the other. If the section is homogeneous in its arguments, then this yields a linear connection.

Using exponential adjunction there are three ways to consider vector fields. Defining vector fields as sections of the tangent bundle of M one can consider them equivalently either as infinitesimal flows, or tangents at the identity map of the microlinear space of diffeomorphisms, i.e., as elements of the Lie algebra of the Lie group Diff(M). The Lie bracket of vector fields can be defined by using this Lie algebra representation and translating the intended geometric meaning directly into algebra. Moreover, using the affine bundle structure of micro squares the Lie bracket can be written in a way resembling the canonical Lie bracket from ring theory. This is because any pair of vector fields determines a micro square in a natural way.

This talk is part of the Synthetic Differential Geometry Seminar series.

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