Which types of differential equations in mathematics are able to broken into the extrinsic normal component?

Question

Which types of differential equations in mathematics are able to broken into the extrinsic normal component like covariant differential $${\mathcal D^2x^a}=d^2x^a+\Gamma^a_{bc}dx^bdx^c$$ where the covariant differential is broken into two parts, the extrinsic normal component $d^2x^a$ and the intrinsic covariant differential component$\Gamma^a_{bc}dx^bdx^c$ (where the $\Gamma^a_{bc}$ is the Christoffel symbol)