Quadrature rules on manifolds play an important role for numerical integration with many applications in science. A quadrature rule on a manifold \( \mathcal M \subset \mathbb R^n \) with \(M\) quadrature points \( \boldsymbol p_1,\dots, \boldsymbol p_M \in \mathcal M \) and associated quadrature weights \( w_1,\dots,w_M \in \mathbb R\) approximates the integral of a continuous function \(f:\mathcal M \to \mathbb R\) over the manifold \(\mathcal M\) by a finite sum, i.e., \[ \int_{\mathcal M} f(\boldsymbol x) \mathrm d \boldsymbol x \approx \sum_{i=1}^M w_i f(\boldsymbol p_i).\]

Of particular importance are Gauß-type quadrature rules which integrate exactly all multivariate polynomials of maximal degree \(N\), i.e., \begin{equation}\label{eq:gauss}\int_{\mathcal M} f(\boldsymbol x) \mathrm d \boldsymbol x = \sum_{i=1}^M w_i f(\boldsymbol p_i), \qquad f(\boldsymbol x)=\prod_{i=1}^n x_i^{\alpha_i}, \; \sum_{i=1}^n \alpha_i \le N. \end{equation} If additionally, all quadrature weights are equal, i.e., \(w_1=\dots=w_M= \frac1M \int_{\mathcal M} \mathrm d \boldsymbol x\), such quadrature rules are of Chebyshev-type. Gauß- and Chebyshev-type quadrature rules are called optimal if for prescribed degree of exactness \(N\) the identity \eqref{eq:gauss} is achieved for the minimal number \(M\) of quadrature points.

The tables below list several quadrature rules for the two-dimensional sphere \[\mathrm S^2 := \left\{ \boldsymbol x \in \mathbb R^3 \;:\; \| \boldsymbol x \|_2 = 1 \right\}\] and the three-dimensional rotation group \[\mathrm{SO(3)}:=\left\{ \boldsymbol R \in \mathbb R^{3\times3} \;:\; \boldsymbol R^\top \boldsymbol R = \boldsymbol I,\; \det \boldsymbol R = 1 \right\}.\] The algorithms for the computation of such quadrature rules together with the general mathematical framework can be found in the PhD Thesis [Gr13Diss].