Deducting from a special case that the electric field transforms as a 2nd rank tensor: Vanishing divergence concludes vanishing vector field?
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Hey dear here is the answer
I assume that your manifold and foliation are smooth and foliation is of codimension 1, otherwise see Jack Lee'a comment. Then pick a Riemannian metric on XX and at each point x∈Mx∈M take unit vector uxux orthogonal to the leaf FxFx through xx: There are two choices, but since your foliation is transversally orientable, you can make a consistent choice of uxux. Then uu is a nonvanishing vector field on XX.
In fact, orientability is irrelevant: Clearly, it suffices to consider the case when XX is connected. Then you can pass to a 2-fold cover X~→XX~→X so that the foliation FF on XXlifts to a transversally oriented foliation on X~X~. See Proposition 3.5.1 of
A. Candel, L. Conlon, "Foliations, I", Springer Verlag, 1999.
You should read this book (and, maybe, its sequel, "Foliations, II") if you want to learn more about foliations.
Then χ(X~)=0χ(X~)=0. Thus, χ(X)=0χ(X)=0 too. Now, recall that a smooth compact connected manifold admits a nonvanishing vector field if and only if it has zero Euler characteristic. Thus, XX itself also admits a nonvanishing vector field.
Incidentally, Bill Thurston proved in 1976 (Annals of Mathematics) that the converse is also true: Zero Euler characteristic for a compact connected manifold implies existence of a smooth codimension 1 foliation. This converse is much harder
Hope its help you
I assume that your manifold and foliation are smooth and foliation is of codimension 1, otherwise see Jack Lee'a comment. Then pick a Riemannian metric on XX and at each point x∈Mx∈M take unit vector uxux orthogonal to the leaf FxFx through xx: There are two choices, but since your foliation is transversally orientable, you can make a consistent choice of uxux. Then uu is a nonvanishing vector field on XX.
In fact, orientability is irrelevant: Clearly, it suffices to consider the case when XX is connected. Then you can pass to a 2-fold cover X~→XX~→X so that the foliation FF on XXlifts to a transversally oriented foliation on X~X~. See Proposition 3.5.1 of
A. Candel, L. Conlon, "Foliations, I", Springer Verlag, 1999.
You should read this book (and, maybe, its sequel, "Foliations, II") if you want to learn more about foliations.
Then χ(X~)=0χ(X~)=0. Thus, χ(X)=0χ(X)=0 too. Now, recall that a smooth compact connected manifold admits a nonvanishing vector field if and only if it has zero Euler characteristic. Thus, XX itself also admits a nonvanishing vector field.
Incidentally, Bill Thurston proved in 1976 (Annals of Mathematics) that the converse is also true: Zero Euler characteristic for a compact connected manifold implies existence of a smooth codimension 1 foliation. This converse is much harder
Hope its help you
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Thus in the primed frame at a given instant the two ends of the object are at x = 0 and x ..... g is called 'the metric' or 'the metric tensor'. .... Special Relativity we use the proper time τ.
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