  The Lanczos-Lovelock models of gravity constitute the most general theories
of gravity in D dimensions which satisfy (a) the principle of of equivalence,
(b) the principle of general co-variance, and (c) have field equations
involving derivatives of the metric tensor only up to second order. The mth
order Lanczos-Lovelock Lagrangian is a polynomial of degree m in the curvature
tensor. The field equations resulting from it become trivial in the critical
dimension $D = 2m$ and the action itself can be written as the integral of an
exterior derivative of an expression involving the vierbeins, in the
differential form language. While these results are well known, there is some
controversy in the literature as to whether the Lanczos-Lovelock Lagrangian
itself can be expressed as a total divergence of quantities built only from the
metric and its derivatives (without using the vierbeins) in $D = 2m$. We settle
this issue by showing that this is indeed possible and provide an algorithm for
its construction. In particular, we demonstrate that, in two dimensions, $R
\sqrt{-g} = \partial_j R^j$ for a doublet of functions $R^j = (R^0,R^1)$ which
depends only on the metric and its first derivatives. We explicitly construct
families of such R^j -s in two dimensions. We also address related questions
regarding the Gauss-Bonnet Lagrangian in $D = 4$. Finally, we demonstrate the
relation between the Chern-Simons form and the mth order Lanczos-Lovelock
Lagrangian.
