Neural networks with a large number of parameters often do not overfit, owing to implicit regularization that favors \lq good\rq{} networks. Other related and puzzling phenomena include properties of flat minima, saddle-to-saddle dynamics, and neuron alignment. To investigate these phenomena, we study the local geometry of deep ReLU neural networks. We show that, for a fixed architecture, as the weights vary, the image of a sample $X$ forms a set whose local dimension changes. The parameter space is partitioned into regions where this local dimension remains constant. The local dimension is invariant under the natural symmetries of ReLU networks (i.e., positive rescalings and neuron permutations). We establish then that the network's geometry induces a regularization, with the local dimension serving as a key measure of regularity. Moreover, we relate the local dimension to a new notion of flatness of minima and to saddle-to-saddle dynamics. For shallow networks, we also show that the local dimension is connected to the number of linear regions perceived by $X$, offering insight into the effects of regularization. This is further supported by experiments and linked to neuron alignment. Our analysis offers, for the first time, a simple and unified geometric explanation that applies to all learning contexts for these phenomena, which are usually studied in isolation. Finally, we explore the practical computation of the local dimension and present experiments on the MNIST dataset, which highlight geometry-induced regularization in this setting.