Lie groups and linear connexions
An r-parameter finite continuous group in the sense of Lie is a set of elements between which a “multiplication” is defined and that. satisfies the following conditions: a) The elements are in one to one correspondence with the points of an N(η α ) in an
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§ 1. Finite continuous groups.
185
remark is not valid if the introduction of a non-symmetric fundamental tensor is combined with a change of dimension. In projective relativity in five coordinates a non-symmetric fundamental tensor does not seem to have been used till yet but in conformal relativity in six coordinates INGRAHAM 1) has recently tried to generalize ScHRÖDINGER's ideas. By transvecting (11 .17 a) wi tli S"" we get (11-33) hence
"' *TA1-']Ä-"'0 [9 *S·!"]Ä.;.
(11.34)
U[9
and according to (4.15) *V91"- *R·V!"Ä .• ;. -- 2"'0 [v *T!-']- 2"'0 [9 *S·!"JA.;. • (11.35) This proves that the condition (11.13} implies that *V.,.. vanishes, that is, that for the connexion *F;;. a covariant constant scalar density field is possible (cf. 111 § 6). From (5 .6) we get according to the condition (11.13) (11.36) and it is easy to show that this is in accordance with (11.32b).
IV.
LIE
groups and linear connexions.
§ 1. Finite continuous groups 2). An r-parameter finite continuous group in the sense of LIE is a set of elements between which a "multiplication" is defined and that satisfies the following conditions: a) The elements are in one to one correspondence with the points of an IJl(r{") in an X, with the coordinates r{"; rx = i, ... , r. b) If the element T'1 belongs to 'YJa. and Tc to r;:z, the product T8 = T'1 Tc belongs to the set and the oa. are analytic functions 3) of the 'YJrr. and r;:-. c) There is an element I called "unity" corresponding to rt such 0 that I T'1 = T'1 I = T'1 for every choice of T'1. L. c. p. 179 footnote 4. General references also for literature: LrE and ENGEL 1888, 1; BIANCHI 1918, 1; EISENHART 1933, 1; VRANCEANU 1947, 1. According to the general character of this book we only consider the "group germ", that means we only are interested in local properties. There is a vast Iiterature on properlies in the large. We mention here only also for literature CARTAN 1927, 2; 1930, 2; 1936, 1; MAYER and T. Y. THOMAS 1935, 1; CHEVALLEY 1946, 1. 3 ) Seefootnote 1 on page 186. 1)
2)
J. A. Schouten, Ricci-Calculus © Springer-Verlag Berlin Heidelberg 1954
186
IV. LIE groups and linear connexions.
d) To every T'1 there belongs an element T"~- 1 such that T"~ T"~- 1 = T"~- 1 T'1 =I. Its Coordinates are analytic functions 1 ) of the rj"". e) (TrJ6)T0 =T'1(T6 T0).
The elements are often transformations in n variables but this is not necessary. From the definition we see that only elements of the group are considered in a neighbourhood of unity. This neighbourhood is called the group germ (Gruppenkeim). We identify each element with its corresponding point in X,. This X, is called group space. To every pair of elements 5, T there belang two elements T 5-1 and 5- 1 T. Two pairs 5,T and 51 ,~ are called
(+ )-equipollent if
T 5_ 1 (- )-equipollent if 5-1 T
= =
7;_ 5;- 1 5! 1 T1 .
From this we get the following deductions: 1. If 5 1 , 7;_; 5 2 , T2 are (±)-equipollent , three of these elements determine the fourth uniquely. 2. If 5 1 ,~; 5 2 , T2 and 5 2 , T2 ; 5 3 , T3 are both (±)-equipo
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