Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 305i.e., M can be reconstructed from A. In particular, the points x of themanifold can be obtained as the 1D irreducible representations x of A,which are all of the form x(f) = f(x). Thus, we can use the algebraof the functions, instead of using the space. In a sense, notices Connes,the algebra is more physical, because we never deal with space–time: wedeal with fields, or coordinates, over space–time. But one can captureRiemannian geometry as well, algebraically. Consider the Hilbert space Hformed by all the spinor fields on a given Riemannian (spin) manifold. LetD be the (curved) Dirac operator, acting on H. We can view A as analgebra of (multiplicative) operators on H. Now, from the triple (H, A, D),which Connes calls ‘spectral triple’, one can reconstruct the Riemannianmanifold. In particular, it is not difficult to see that the distance betweentwo points x and y can be obtained from these data byd(x, y) = sup {f∈A,||D,f||