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We use vielbein bundle's horizontal lift path integral formulation and gaugetheory's holonomy map to compactly describe parallel transport and geodesicequations on a manifold. This is first applied to the geometry of generalrelativistic Schwarzschild as a review. The Newton-Cartan take of theSchwarzschild metric derived by previous literature is then adopted, and theanalysis is repeated for both non-torsional and torsional geometry. Transportcurves considered include a circular timeless loop, circular geodesic loop,radial geodesic loop, and a stationary loop. The findings on the threegeometries are then contrasted, key differences summarized and the differingmechanisms discussed. In particular, we find a "performance tradeoff" betweenthe torsional and non-torsional Newton-Cartan theory: the former yields moreaccurate equations of motion whereas the latter simulates relativisticdilation/contraction effects to an exact degree.

Path Integral Formulation and Holonomy in Newton-Cartan Schwarzschild Geometry