  When analyzing modern machine learning algorithms, we may need to handle
kernel density estimation (KDE) with intricate kernels that are not designed by
the user and might even be irregular and asymmetric. To handle this emerging
challenge, we provide a strong uniform consistency result with the $L^\infty$
convergence rate for KDE on Riemannian manifolds with Riemann integrable
kernels (in the ambient Euclidean space). We also provide an $L^1$ consistency
result for kernel density estimation on Riemannian manifolds with Lebesgue
integrable kernels. The isotropic kernels considered in this paper are
different from the kernels in the Vapnik-Chervonenkis class that are frequently
considered in statistics society. We illustrate the difference when we apply
them to estimate the probability density function. Moreover, we elaborate the
delicate difference when the kernel is designed on the intrinsic manifold and
on the ambient Euclidian space, both might be encountered in practice. At last,
we prove the necessary and sufficient condition for an isotropic kernel to be
Riemann integrable on a submanifold in the Euclidean space.
