  We construct in this article a class of closed semi-bounded quadratic forms
on the space of square integrable functions over a smooth Riemannian manifold
with smooth boundary. Each of these quadratic forms specifies a semi-bounded
self-adjoint extension of the Laplace-Beltrami operator. These quadratic forms
are based on the Lagrange boundary form on the manifold and a family of domains
parametrized by a suitable class of unitary operators on the boundary that will
be called admissible. The corresponding quadratic forms are semi-bounded below
and closable. Finally, the representing operators correspond to semi-bounded
self-adjoint extensions of the Laplace-Beltrami operator. This family of
extensions is compared with results existing in the literature and various
examples and applications are discussed.
