# How do 'locally Euclidean' and 'Lorentzian' requirements in manifolds reconcile?

In GR, we define our manifolds to be locally Euclidean. However, we also demand that our metric tensor have a Lorentzian signature. Since the metric tensor is a measure of curvature, doesn't the first property imply that our metric should be locally Eulcidean (Riemannian)? How does imposing a Lorentzian signature on the metric preserve the locally Euclidean property of the manifold?

Manifolds are by definition locally Euclidean (i.e. locally look like $$\mathbb{R}^n$$, the $$n$$ dimensional Euclidean space). There is no metric given a priori.