Extending traditional lattice field theory beyond flat space to a
smooth Riemann manifolds requires a Quantum extension of Finite
Elements (QFE) compatible with the discrete quantum geometry of a
``triangulated'' simplicial complex. Consideration of ultraviolet
counter terms, the affine projection between flat and curved space are
developed and tested in the context of the 2d and 3d Ising conformal
field theory (CFT) on spheres and cylinders. Challenges for QFE to
theories with fermionic and gauge fields will be discussed.