A cube has a constant positive curvature what seems to be connected to some Lie algebra structure constants, and torsion is involved in the second cohomology group H2(G, U(1)). Well, sometimes it's said that not the manifold but rather the connection has got a torsion, that's to say the torsion tensor depends on a connection. The abstract manifolds department puts it this way (here X and Y are some vector fields):
The curvature is defined as follows:
Those equations have counterparts in tensor formulation and as well in the context of forms (that means homogeneous polynomials). In the latter case they are also called Cartan's first (relating the Lie derivative to the inner and exterior derivative) and second structure theorem.
The global Gauss-Bonnet theorem expresses the Euler-Poincaré characteristic of a manifold as a curvature integral. And it's related to the Riemann-Hurwitz formula.
Weierstrass, Teichmüller, Hermitian metric, Kähler spaces and so on have perhaps as well to do with our project. And those Willmore surfaces look interesting as well.
What about s.th. like vector analysis (scalar and vector fields, div rot=0, grad, Laplace-Beltrami, tensors, forms)?