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We show that in the presence of $n$-fold rotation symmetries andtime-reversal symmetry, the number of fermion flavors must be a multiple of$2n$ ($n=2,3,4,6$) on two-dimensional lattices, a stronger version of thewell-known fermion doubling theorem in the presence of only time-reversalsymmetry. The violation of the multiplication theorems indicates anomalies, andmay only occur on the surface of new classes of topological crystallineinsulators. Put on a cylinder, these states have $n$ Dirac cones on the top andon the bottom surfaces, connected by $n$ helical edge modes on the sidesurface.

Rotation Anomaly and Topological Crystalline Insulators