We compute conformal anomalies for conformal field theories with free conformal scalars and massless spin $1/2$ fields in hyperbolic space $mathbb{H}^d$ and in the ball $mathbb{B}^d$, for $2leq dleq 7$. These spaces are related by a conformal transformation. In even dimensional spaces, the conformal anomalies on $mathbb{H}^{2n}$ and $mathbb{B}^{2n}$ are shown to be identical. In odd dimensional spaces, the conformal anomaly on $mathbb{B}^{2n+1}$ comes from a boundary contribution, which exactly coincides with that of $mathbb{H}^{2n+1}$ provided one identifies the UV short-distance cutoff on $mathbb{B}^{2n+1}$ with the inverse large distance IR cutoff on $mathbb{H}^{2n+1}$, just as prescribed by the conformal map. As an application, we determine, for the first time, the conformal anomaly coefficients multiplying the Euler characteristic of the boundary for scalars and half-spin fields with various boundary conditions in $d=5$ and $d=7$.
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