I'd like to know the definition of "tangential component" in this case, it is question 3 of page 57 of Do Carmo's book Riemannian Geometry:
It says:
Define $\nabla_XY(p) = $ tangential component of $\overline\nabla_{\overline X}\overline Y(p) $, where $\overline\nabla$ is the Riemannian connection of $\overline M$.