Local flatness at a point, vanishing curvature and global Minkowski geometry are three different statements

This note separates three statements that are frequently conflated, and records their exact status in this library. The local time-orientation reduction below checks the domains of its cited suppliers.

• First-order flatness at a point is automatic. Every smooth Lorentzian four-manifold has point-normal and exponential charts around each point. To apply the cited time-oriented supplier at $p$, choose a timelike $v\in T_{p}M$ and extend its components constantly in any initial chart. Smoothness of $g$ and $g_p(v,v)<0$ make this field timelike on a smaller open neighborhood $V\ni p$. Thus $(V,g{\mathrm{\mid }}_V,N)$ is time-oriented and meets the supplier's hypotheses. The metric-selected connection restricts to that of $g{\mathrm{\mid }}_V$ by its uniqueness, and local geodesic solutions and their exponential charts agree with the ambient ones by uniqueness of the same coordinate differential equation. Hence the supplied charts have $g_{ab}(p)={\eta }_{ab}$, ${\mathrm{\partial }}_a{g}_{bc}(p)=0$ and ${\mathrm{\Gamma }}^a{}_{bc}(p)=0$; the construction is first order and makes no claim about second derivatives of the metric (Point-normal and exponential coordinate charts of the metric-selected connection clauses (a)-(b)). Consequently the existence of locally inertial coordinates at an event is a property of every smooth Lorentzian metric, not an indication of vanishing curvature, and by itself it is not the statement that any neighborhood is Minkowski space.

• The curvature is the second-order obstruction. The point-normal formula of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (c) expresses the curvature at the point through the second derivatives of the metric in any point-normal chart, and the curvature is a tensor field, so its vanishing is chart-independent (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). If the curvature vanishes on a normal neighborhood, the exponential chart is flat to all orders. With the normal domain $W\subseteq T_{p}M$ and $U={\exp }_p(W)$, the supplier gives the isometry ${\exp }_p:(W,g_p)\to (U,g)$. For the coordinate isomorphism $\psi :T_{p}M\to {\mathbb{R}}^4$ of a $g_p$-orthonormal basis, the map $u=\psi \circ {\exp }_p^{-1}:(U,g)\to (\psi (W),\eta )$ is an isometry onto an open subset of Minkowski space (Vanishing curvature on a normal neighborhood is a local isometry to Minkowski space clauses (a)-(d)). This is the exact sense in which a flat metric is locally Minkowski.

• Global Minkowski geometry is strictly stronger. Vanishing curvature on a manifold does not determine its topology, its holonomy or its causal structure. The companion examples page exhibits a flat Lorentzian quotient of Minkowski space by a translation whose time coordinate is periodic: the metric is flat, the Einstein tensor vanishes, yet the manifold is not diffeomorphic to ${\mathbb{R}}^4$, admits closed timelike curves, and carries no global inertial chart; the theorem above is local precisely because it uses a normal neighborhood, not the whole manifold. Statements such as "the flat model is the spacetime" therefore require additional global hypotheses (simple connectedness, completeness, injectivity of the exponential map, a global chart) that are not part of the local computations of this page.

• The flat model and its Killing structure. In the model of The flat model spacetime: the Minkowski metric on R^4 and its inertial chart the coordinate fields ${\mathrm{\partial }}_0,\dots ,{\mathrm{\partial }}_3$ are global Killing fields and the geodesics are complete straight lines (The flat model: vanishing connection and curvature, translation Killing fields and straight geodesics clauses (c)-(d)). A manifold merely locally isometric to this model need not inherit any of these global structures: local isometries do not glue to global ones without monodromy and completeness hypotheses. This is why the physical and mathematical statements of this page are phrased neighborhood by neighborhood.

• Scope. No classification of flat Lorentzian four-manifolds and no global rigidity statement is made or used on this page. The note only fixes which of the three statements above each item of the page proves, and warns that the words "locally flat", "curvature-free" and "Minkowski" are not interchangeable.