Differential geometry can contain different ranges. If it is differential geometry in the second year of mathematics department, you may refer to curve theory and surface theory in three-dimensional Euclidean space. It is enough to learn linear algebra and multivariate calculus, and it is best to take an examination of spatial analytic geometry. A little more advanced, it is probably equivalent to elementary Riemannian geometry, and needs some foundations of differential manifolds. If you are an introduction to the branches of mathematics, including the connections on fiber bundles, you may need some knowledge of algebraic topology and Lie groups, preferably having studied differential equations.

But if it is modern differential geometry, it needs more. First of all, linear algebra, but this time it's multilinear algebra, otherwise you won't understand what tensor is. Find an advanced linear algebra (algebra is a good introduction). Then there is topology, otherwise you will not understand what manifold is, and the requirements for topology will not be too high. Seeing homology is almost enough. Then lie algebra, and then Chen Shengshen's lectures on differential geometry.