By Y Matsushima

Long ago 30 years, differential geometry has gone through an immense swap with infusion of topology, Lie idea, complicated research, algebraic geometry and partial differential equations. Professor Matsushima performed a number one position during this transformation through bringing new ideas of Lie teams and Lie algebras into the examine of actual and complicated manifolds. This quantity is a set of the entire forty six papers written via him.

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**Example text**

Let 5R, be the radical of the derived algebra 2 ' of 2. Then 3ft, is composed only of nilpotent matrices. Let % be the space of r-times contravariant and stimes covariant tensors and 9! the totality of tensor invariants of 3t, in % ,,. Since 3t is an ideal of 2, 9c ,, is invariant under 8. Let i 0 l 1 l Q l 4 t tt t ( rs r a r ( r where direct summations are extended to sufficiently high orders of r, s, O is an invariant space under 8 and we denote by (2) the representation of 2 induced in Q .

Z ai Brt r qv 0ii u K (~ %< - • • • - r )E. r

Of the eigen-space 2 . =*yi 0 = 1. m) r (8) lA,Mt)— «%-H*«+i \A u, rs ni = (/'*»& « = 1 . 2, .... ) 0 = 1 , 2, . . e%.. implies B„F=BJF. Since ^ , F . = A , / F = -£KF. I We re- 32 46 Y. We have MATSi/siriMA. I V o I . 23, A, F. = *F.. S From this we see easily that F„ has the following form : Since B is evidently a replica of A, , we have from (5), (7) and (8) the following equations : n s ,M0 =-m (*=T, 2, —MH0 t ! 5 , ^ ( 0 = i ' « , j - f - 7 ( j + ^ ( j * t + ••• + 7r»j&(/», { BJF. g) = vF„, F , a .