We study the quantum geometry of the fuzzy sphere defined as the angular
momentum algebra [xi,xj]=2λpϵijkxk modulo setting
∑ixi2 to a constant, using a recently introduced 3D rotationally
invariant differential structure. Metrics are given by symmetric 3×3
matrices g and we show that for each metric there is a unique quantum
Levi-Civita connection with constant coefficients, with scalar curvature 21(Tr(g2)−21Tr(g)2)/det(g). As an
application, we construct Euclidean quantum gravity on the fuzzy unit sphere.
We also calculate the charge 1 monopole for the 3D differential structure.Comment: 15 pages latex, 1 figur