dually-flat.md

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Dually flat manifolds

Let $$(M,g,\nabla,\nabla^)$$ be a dually connected Riemannian manifold. It will be called a dually flat manifold if $$\nabla$$ is flat (and consequentially $$\nabla^$$ too).

If $$M$$ is dually flat, then there exists locally two affine coordinate systems $$\theta=(\theta^1,\dots,\theta^n)$$ and $$\eta=(\eta_1,\dots,\eta_n)$$ for $$\nabla$$ and $$\nabla^$$ respectively, with local basis $$\partial_i \coloneqq \frac{\partial}{\partial \theta^i}$$ and $$\partial^j \coloneqq \frac{\partial}{\partial \eta_j}$$, which are affine in the sense that $$\nabla_{\partial_i} \partial_j \equiv 0$$ and $$\nabla^_{\partial^i} \partial^j \equiv 0$$. In other words, $$\theta$$ vanishes the Christoffel symbols of $$\nabla$$ and $$\eta$$ vanishes the Christoffel symbols of $$\nabla^*$$.

In the case where $$\nabla$$ and $$\nabla^*$$ are duals, we can construct such coordinate systems $$\theta$$ and $$\eta$$ with the additional property that their local frames are duals:

$$ \langle \partial_i, \partial^j \rangle = \langle \partial^i, \partial_j \rangle = \delta_{ij}.$$

Such coordinate systems are called dual coordinate systems for $$(M,g,\nabla,\nabla^*)$$.

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