Given an abelian scheme over a smooth curve over $\overline{\mathbb{Q}}$, we can associate two height functions: the fiberwise defined Neron-Tate height and a height function on the set of algebraic points of the base curve. For any irreducible subvariety $X$ of this abelian scheme, we prove that the Neron-Tate height of any algebraic point in an explicit Zariski open subset of $X$ can be uniformly bounded from below by the height of its projection to the curve. We use this height inequality to prove the Geometric Bogomolov Conjecture over characteristic $0$. This is joint work with Philipp Habegger.