# The exterior connections

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Let $M$ be a manifold with a fiber bundle $E$ which is a modulus over the exterior forms. An exterior connection is defined by the formulas:

$$(\alpha \wedge \beta).s= \alpha .(\beta .s)$$

$$\nabla (\alpha .s)=d \alpha .s +(-1)^{deg(\alpha)} \alpha . \nabla (s)$$

with $\alpha ,\beta \in \Lambda^* (TM)$, and $s$ a section of $E$. The curvature is $R=\nabla \circ \nabla$ such that:

$$R(\alpha .s)=\alpha . R(s)$$

Can we make differential geometry with the exterior connections?

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