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 Dec 10 comment Basis of differential one-form confusion This, as it is, has nothing to do with general relativity. It's just a very basic question in differential geometry. Oct 11 comment Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat? The volume of the object with 5 points as vertices will be zero (that depends only on the dimension), but the space sill not be flat. A more visual example would be a curved surface and four points on it. Oct 11 comment Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat? @user12262: Take the same example and change the usual metric to something else, then the space will not be flat (non-zero curvature) and the determinant of any 5 points will vanish (it is the same, no change here). Oct 10 comment Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat? @user12262: I've already given you an example. Take 3d Euclidean space, it is flat (has zero curvature). On the other hand you can take four point, the vertices of a pyramid, they are not flat in a plane. If one is flat and the other is not, obviously the two notions cannot be the same. Oct 10 comment Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat? You still don't get it. Flatness of the spacetime (no matter what dimension) and flatness as configuration of points in a spacetime are two different things that you confuse. Oct 9 comment Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat? @user12262: Your question is about how to determine if a given space-time is flat or not. The box in the book you referred to talks about whether a number of points lie flat in a plane or not. These are two different things that you are confusing. And it is obvious, because one is about the space-time the other is about a set of points. Take Euclidean space, it is flat i.e. has zero curvature, yet you can have four points in it that do not lie flat in a plane, right? Oct 8 comment Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat? @user12262: Obviously the word "flat" here is used in a different context and meaning. It doesn't related to the curvature of the space. It expresses a relation between point, whether they belong in a common plane and so on. Oct 7 awarded Commentator Oct 7 comment Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat? @user12262: Four points in 3D Euclidean space can be in a 2D plane or not, to tell you can compute the volume of the tetrahedron the form, if it is zero they lie in a plane, if not they don't. That is something completely different from what the curvature of the space is. Oct 7 comment Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat? @user12262: That is not related to flatness. Oct 7 comment Given a spacetime in terms of values of Lorentzian distance, how to determine whether it is flat? @user12262: Where exactly in box 13.1 is the part about flatness? Nov 7 comment (Strong) causality condition What he proves is not that a compact set has closed causal curves, but that a compact space-time has closed causal curves. Take Mikowski space-time and a compact subset, it will not have closed causal curves. May 11 comment How to proof Frobenius Theorem in general? The case k=1 follows from an existence and uniqueness theorem for ODE's. The step to k=2 from the proof you know should be the same as the on from k-1 to k in the induction. Apr 20 comment Link between a topological space and a manifold This doesn't make sense! The manifold $M$ comes with a topology i.e. with the collection of subsets $T$. Apr 4 comment Can we define a induced metric like this? The induced metric is just the restriction of $g$ to the tangent space of the submanifold. Or the restriction of $h$. The $h$, if you raise one index, is the projection operator. Mar 10 awarded Supporter