Narrow Results

Given a principal $G$-bundle $P\to M$ and two $C^1$ curves in $M$ with coinciding endpoints, we say that the two curves are holonomically equivalent if the parallel transport along them is identical for any smooth connection on $P$. The main result in this paper is that if $G$ is semi-simple, then the two curves are holonomically equivalent if and only if there is a thin, i.e. of rank at most one, $C^1$ homotopy linking them. Additionally, it is also demonstrated that this is equivalent to the factorizability through a tree of the loop formed from the two curves and to the reducibility of a certain transfinite word associated to this loop. The curves are not assumed to be regular.

Some families of H-valued vector fields with calculable Lie brackets are given. These provide examples of vector fields on path spaces with a divergence and we show that versions of Bismut type formulae for forms on a compact Riemannian manifold arise as projections of the infinite dimensional theory.

Let M be a C^∞ manifold, and let be the space of all smooth maps from [0, 1] to M. We investigate geometric structures on constructed from the geometric structures on M. In particular, we show that a generalized (almost) complex structure on M produce a generalized (almost) complex structure on .

Let M be a complex manifold and let PM ≔ C^∞([0, 1], M) be space of smooth paths over M. We prove that the induced almost complex structure on PM is weak integrable by extending the result of Indranil Biswas and Saikat Chatterjee of [Geometric structures on path spaces, Int. J. Geom. Meth. Mod. Phys.8(7) (2011) 1553–1569]. Further we prove that if M is smooth manifold with corner and N is any complex manifold then induced almost complex structure 𝔍 on Fréchet manifold C^∞(M, N) is weak integrable.

Given a compact symplectic manifold M, with integral symplectic form, we prequantize a certain class of functions on the path space for M. The functions in question are induced by functions on M. We apply our construction to study the symplectic structure on the solution space of Klein–Gordon equation.

Let $M$ be a Riemannian manifold and ${𝒫}M$ be the space of all smooth paths on $M$. We describe geodesics on path space ${𝒫}M$. Normal neighborhoods on ${𝒫}M$ have been discussed. We identify paths on $M$ under “back-track” equivalence. Under this identification, we show that if $M$ is complete, then geodesics on the path space yield a double category. This double category has a natural interpretation in terms of the worldsheets generated by freely moving (without any external force) strings.