Linear algebra nugget of the day. You can classify a 2d quadratic form without finding eigenvalues. Det tells you if the eigenvalues are the same sign and if det is positive, trace tells you which sign. Exercise: what if det is 0?
I completely agree that it is wise to remain at least dimly aware of how the many geometrical structures present in each solution of the Einstein equation depend upon the spacetime metric (just as it is wise to understand Euclidean geometry as a thing of many interrelated parts.)
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This is a point first championed by Hermann Weyl, and subsequently elaborated and deepened by Ehlers-Pirani-Schild and Trautman (and, ultimately, N M J Woodhouse.) I have long wished that it were far more widely and fully understood and appreciated.
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