Holmstrom Torsion in differential geometry

Ray and Singer: R-torsion and the Laplacian on Riemannian mfds. Adv Math 1971

http://ncatlab.org/nlab/show/torsion mentions several notions of torsion.

http://mathoverflow.net/questions/20493/what-is-torsion-in-differential-geometry-intuitively

http://mathoverflow.net/questions/58471/analytic-torsion-in-the-derived-category

nLab page on Torsion in differential geometry