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

Structure of Lanczos-Lovelock Lagrangians in Critical Dimensions