# What is the variation of Gauss-Bonnet term a total derivative of?

What is the variation of Gauss-Bonnet term total derivative of?

i.e. Variation of Gauss-Bonnet combination $= \nabla_{\mu} C^{\mu}$.

What's $C^{\mu}$ in 4-dimensions?

• A fairly careful and recent discussion of this can be found here: arxiv.org/abs/1008.5154v1. They discuss this using the language of forms (where the result has been known for a long time), and using just functions of the metric. – Lloyd Sep 19 '17 at 14:17

According to this website, for a four dimensional manifold, $$G = \nabla_{\alpha}J^{\alpha},$$ where $$G = R^2 -4 R_{\alpha \beta} R^{\alpha \beta} + R_{\alpha \beta \gamma \delta}R^{\alpha \beta \gamma \delta},$$ and $$J^{\alpha} = \epsilon^{\alpha \beta \gamma \delta} \epsilon_{\rho \sigma}^{\;\;\; \mu \nu} \Gamma^{\rho}_{\;\; \mu \beta} \left[ \frac{1}{2} R^{\sigma}_{\;\; \nu \gamma \delta} + \frac{1}{3} \Gamma^{\sigma}_{\;\; \lambda \gamma} \Gamma^{\lambda}_{\;\; \nu \sigma} \right].$$ So $G$ becomes a topological term in the action, which does not contribute to the dynamics. However, I have yet to check it myself...