# A Course of Differential Geometry by Campbell J.E.

By Campbell J.E.

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Example text

N, when applied to any tensor components, generate other tensor components. MS Let L.... and assume that jT** We ... >*',.. have which we '" ^ >jv JL a', b',... briefly write 2 T/ X'... are tensor components. = JLT*'^~' MN " IV a, b, ... = TJUN. 2) ))jf. T=--T ^X Now ^\ i We a have written x o T* -rJQ 3* , . . t has the upper integers a, 6, ... ) and that the upper integers in note that Jlf M the same as the lower integers in T. Similarly we note that the lower integers in N are the upper integers in T.

Is the general and the simplest case when Finally functions both not A are and not a constant, and A (K) 2 (K) of K. 29. Conditions K we have In this case we have two invariants, say u and v. 2) be respectively the same functions of may u and v. We now know in all cases the tests which will determine whether two assigned ground forms are, or are not, equivalent. The functions 30. of the When called rotation functions. measure of curvature * is constant ground form we saw the 24] that the re- [ form depends on finding an integral of the complete system of differential duction equations 0.

U Here the . ri have no meaning of suffixes differentiation or of being tensor components. 5) Jl ), and we can express the ground form in terms of the in- variants. we can say that the necessary and sufficient that conditions that two ground forms may be equivalent are that for each is, transformable the one into the other In this case form the equations A 0'i Uk) = 0toK---'M'fi) ( 17 - 6) be the same. may For special forms of the ground form we may not be able Thus to find the required n invariants to apply this method.