  In this paper, we study the Radial Basis Function (RBF) approximation to
differential operators on smooth tensor fields defined on closed Riemannian
submanifolds of Euclidean space, identified by randomly sampled point cloud
data. {The formulation in this paper leverages a fundamental fact that the
covariant derivative on a submanifold is the projection of the directional
derivative in the ambient Euclidean space onto the tangent space of the
submanifold. To differentiate a test function (or vector field) on the
submanifold with respect to the Euclidean metric, the RBF interpolation is
applied to extend the function (or vector field) in the ambient Euclidean
space. When the manifolds are unknown, we develop an improved second-order
local SVD technique for estimating local tangent spaces on the manifold. When
the classical pointwise non-symmetric RBF formulation is used to solve
Laplacian eigenvalue problems, we found that while accurate estimation of the
leading spectra can be obtained with large enough data, such an approximation
often produces irrelevant complex-valued spectra (or pollution) as the true
spectra are real-valued and positive. To avoid such an issue,} we introduce a
symmetric RBF discrete approximation of the Laplacians induced by a weak
formulation on appropriate Hilbert spaces. Unlike the non-symmetric
approximation, this formulation guarantees non-negative real-valued spectra and
the orthogonality of the eigenvectors. Theoretically, we establish the
convergence of the eigenpairs of both the Laplace-Beltrami operator and Bochner
Laplacian {for the symmetric formulation} in the limit of large data with
convergence rates. Numerically, we provide supporting examples for
approximations of the Laplace-Beltrami operator and various vector Laplacians,
including the Bochner, Hodge, and Lichnerowicz Laplacians.
