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We discuss a general fourth-order theory of gravity on the brane. In general,the formulation of the junction conditions (except for Euler characteristicssuch as Gauss-Bonnet term) leads to the higher powers of the delta function andrequires regularization. We suggest the way to avoid such a problem by imposingthe metric and its first derivative to be regular at the brane, while thesecond derivative to have a kink, the third derivative of the metric to have astep function discontinuity, and no sooner as the fourth derivative of themetric to give the delta function contribution to the field equations.Alternatively, we discuss the reduction of the fourth-order gravity to thesecond-order theory by introducing an extra tensor field. We formulate theappropriate junction conditions on the brane. We prove the equivalence of boththeories. In particular, we prove the equivalence of the junction conditionswith different assumptions related to the continuity of the metric along thebrane.

Generalized Israel Junction Conditions for a Fourth-Order Brane World