# 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?

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## 2 Answers

If you just want to know why the Gauss-Bonnet Term is topological, you should take a look at the generalized gauss bonet theorem.

The integral over the gauss-bonet term is proportional to the euler-characteristic, which is a topological invariant, so it can't contribute to the dynamics.

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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...

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