SOME INVARIANT THEOREMS ON GEOMETRY OF EINSTEIN NON-SYMMETRIC FIELD THEORY

This paper generalizes Einstein's theorem. It is shown that under the 
transformation 
TΛ:Uikl→U¯ikl≡Uikl

It is still shown that for arbitrary U, the transformation that makes curvature i tensor Skm(U (or Ricci tensor Uik ik ik must be T A transformation, where V (its components are Vik is a second order differ- entiable covariant tensor field with vector value.

i. INTRODUCTION.
When A. Einstein devoted himself to research on relativism in his symmet- tic field , he regarded non-symmetric gij (or gij) and non-symmetric affine con- nection D (its coefficinets are F i ik in local coordinates {x }) as independent variables such that the number of independent variables increased from 50 (gij and r e ij g rik) ik are all symmetric for lower coordinates) to 80 (16 gij or g and 64 With so many covariant variables, it was impossible to choose them according to the principle of relativism alone.To overcome this difficulty, Einstein introduced a very important concept, transposition invariance.This "transposition invariance" (or transposition symmetry) meant that when all Aik were transposed (AikT Aki) all equations were still applicable [2] Einstein supposed that field equations were transposition invariant.He thought that in physics this hypothesis was equivalent to the law that positive and negative electricity occurred symmetrically.

F
As the Ricci tensor Rik(F represented by connected coefficients ik was not transposition invariant, Einstein introduced a "pseudo-tensor" U ik instead [3] its definition was Denoting F ik by Uik we obtained Fik--Uikitk Then the Ricci curvature denoted by U was Rik--Uik, s it sk + is tk--Sik Sik" Einstein proved that Sik were transposition invariant and the following.

THEOREM (EINSTEIN). [i] Under the transformation
the Ricci tensor Sik of U is invariant; i.e., under the transformation (4), there are Sk Sik for arbitrary U where Sik--Sik(U ).In (4) X and is a differ-'J xJ entiable function on a manifold M. REMARK.Einstein gave transformation (1.4) for n=4, but we will still call trans- formation (1.4) the Einstein transformation for general n (_ 2) One asks naturally, how about the converse of Einstein's theorem? A. Einstein and B. Kaufman did not solve the problem.It has remained unsolved.
In this paper, we generalize Einstein and Kaufman's results to an arbitrary n (K 2) dimensional manifold M. Objects which we discuss are not limited to the Ricci tensor i Sik of U.Besides Sik, we discuss curvature tensor Skzm and scalar curvature S.
Then, for general n ( 2), we give some invariant theorems on curvature tensor i Skzm, Ricci tensor Sik and scalar curvature S of U.For this, first we generalize Einstein's transformation.Finally, we give converse theorems of theorems for arbitrary n (>_ 2).
These are the main results of this paper.In the special case n=4, we answer the problem abovementioned; that is, a converse to Einstein's theorem.To establish expressions for the curvature tensor Sikm' Ricci tensor Sik and scalar curvature S of U for arbitrary n ( 2), it is necessary to give transformation between U and r for arbitrary n (>_ 2).
In (1.1), let =k, add from 1 to n obtaining t F t Uit it-nFt t (n-l) Fit.
(2.3) Substituting (2.5) into (1.1),we can solve g g 3), and definitions we obtain immediately i PROPOSITION 1. Curvature tensor Skim, Ricci tensor Sik and scalar curvature S of U are respectively -n---[ 6m st Uk n 1 Ukt m (5) Sg Rik g ik,s g it sk + g is Utk S(U) (2.7) When n>_ 2, it is not difficult to verify that Ricci tensor S ik and scalar curvature S of U are transposition invariant.
THEOREM i. Curvature tensor S ikm, Ricci tensor Sik and scalar curvature S of U are all invariant under the following transformation z Uik Uik Uik + i Ak 8k Ai' whexe A A.d is a closed 1-differential form on a manifold M; i.e., d A 0.
REMARK.The transformation (2.8) is a generalization of Einstein's transformation (1.4).In fact, as is a closed 1-differential form on a manifold M (d, 0), then by the Poincare Lemma, there exists a coordinate neighborhood MI=M and a different- ---(k=l n) Therefore in a local iable function X such thatA k x k k neighborhood, for example M1, the transformation (2.8) conforms with Einstein's trans- formation (1.4)Because an exact differential form dX is a closed differential form (d 0), T x is a transformation which makes Skzm,i Sik and S invariant.When n=4, Einstein's theorem is a special case of the above theorem I.
PROOF.In local coordinates {xi}, let A. X., then Consider the transformation U Z T Uik ik ik / igk i (2.9) where is a 1-differential form, in local coordinates {xi}, .dxj.
Having the above result for theorem I, we ask naturally if the transformation i which makes curvature tensor SkZm (or Ricci tensor Sik invariant is the transfor- mation T? For this, although we cannot give the complete answer it is a very difficult problem we have the following results.i THEOREM 2. The transformation that makes curvature tensor Skzm Sik) of some U invariant T U 6i ik ik Uik + gk must be where =.ck is a 1-differential form.
--i S i PROOF.Similar to the proof of theorem I, we obtain Skm km(U) -6Zs) (Ukm k m i(ut 5 s (U L + kO.m n-I 6 st + 6 fit n---(Ukt + t tgk (Um + 6 m) (6 gt,mn mk + 1 3iut (2.10) Now we give the converse of theorem i.For this, what we must emphasize is that because of theorem i, the transformation T A makes curvature tensor S ikm and Ricci tensor Sik of every U invariant.
The following theorems, 4 and 5 respectively, are the converses f theorem 1 on i curvature tensor Skm and Ricci tensor Sik.
THEOREM 4. Let V be a second order differentiable covariant tensor field with vector value and its components be V ik in local coordinates {x i }.If the transfor- of every U invariant then it implies makes curvature tensor Skm Vik i k i whereA----A.dx is a closed 1-differential form; i.e.T must be T A.  Since-iSkm Skmi (for every U) now km(U) 0 (i, k, , m I, n and for every U).Therefore,

iand
Ricci tensor Sik of U, first To give the definitions for curvature tensor Skzm i andRicci tensor Rik of let us give reasonable definitions for curvature tensor Rkm connection D (F z ik (order of lower coordinates is very important; what we give here differs by a minus sign from what is sometimes used, for example, in Pauli's relativism).