Let \(u\) and \(v\) be smooth functions on the sphere \(\mathbb{S}^d_R\) with radius \(R>0\). Then using (0x67dbf0ac) implies

\[ \langle \nabla u, \nabla v\rangle_{L^2(\mathfrak{X}(\mathbb{S}^d_R))}=\langle u,-\Delta v\rangle_{L^2(\mathbb{S}^d_R)}. \]

See also Link to heading

• Bernstein Inequality on the sphere with covariant derivatives