Tensor Analysis

Differential Parallelism.

A fruitful contribution to tensor analysis, due to Levi-Civita (1917), is the concept of differential parallelism. The metrical 5„ can always be imagined as hyper surface embedded in a Euclidean space E, of ln(n+ i) dimensions. 0(x,) and O'(x,-Edx,) being points of S„, and OP a Civita transfers OP by a Euclidean parallel shift to and defines O'P', the orthogonal projection of 0' upon the E-plane tangential to S„, at 0, as parallel to OP. In spite of the ultra spacial construction, parallelism is intrinsic, expressible by the alone. If A' be the components of OP, and A'-}-6A' those of 0'P', = A.dx, (7) For covariant components, (MK= a, (.1 A,dxx. These formulae give, in any co-ordinates, the changes of the components of a vector produced by its translation or parallel shift through dxx.

In local geodesic co-ordinates (g. stationary at 0), o. Alternatively, after Weyl, parallelism may be defined in the non metrical by putting 3A'= —N.A.Kclxx, where N. = r,,`. are freely prescribable functions of position, fixing the affine con nection of the otherwise amorphous manifold. One can continue to hold this more general, affine view in developing further con sequences, as of late was done by many writers; but this subject cannot here be dwelled upon. If metrics are impressed upon the affine manifold, the requirement that the size of a vector shall remain unaltered by a translation, A") = o, gives r,:.= {Kx,c}, leading again to (7).

The parallel shift is a powerful means of obtaining differential tensors. Being the difference of not copunctal vectors, 8A,, is not a vector; but, if A. be considered as a field, its value at 0' ,(3,4K is AK+ while the vector transferred from 0 to 0' is x the sufficiency of this condition for the reducibility of gudlx,dx. to a form with constant coefficients was proved by Lipschitz as early as 1869.

Riemann's

Symbols and mann's own set of symbols is the covariant tensor (tg, Arc) = Rwx,c=gmed??,.x. Conversely, Rcl...x = gAa(tp,, AK). Like (9), the symbols are antisymmetric in K, X. Three more linear relations hold between them. This reduces the number of in dependent symbols to I) ; e.g., six for a three-space, twenty for an S4, and but one for a surface, say (12, 12) . This symbol divided by g is an invariant of the surface, its Gaussian curvature (K.) (See DIFFERENTIAL GEOMETRY.) Formula (8) gives for the rotation of a two-vector carried around a surface-element (of area =1 go"), A 0 = Kda; the vector on its return overshoots its original direction or falls short of it by IKIda according as K o. The Gaussian curvature thus appears as the rotation per unit area. As a consequence, the excess (over 7r) of the angle sum in any geodesic triangle is the "total curvature" of the triangle, a famous theorem due to Gauss.

In the case of three or more dimensions the curvature proper ties can no longer be expressed by a single magnitude, but require for their description the knowledge of the whole curvature tensor or the associated Riemann symbols. The concept of Gaussian

curvature is now replaced by that of Riemannian curvature. This is, at any point 0(x) of the set of Gaussian curvatures Kp of geodesic surfaces of all possible orientations laid through 0. If haft be the metrical tensor of such a surface as sub-manifold, then (12, 12)h, the symbol to be calculated with //co. This and the determinant h can be expressed in terms of the tensor of the manifold and the vector pair, dEL, dm, fixing the orienta tion. The result is is again antisymmetric in a, j3. Therefore, and since AX" is a vector, Kafris a mixed tensor. This is the or curvature tensor. Its meaning is best expressed by (8) itself, which reads : the change of a vector carried around an oriented surface-element is half the inner product of the curvature tensor into that element and the vector. An alternative deduction of Rf",,x can be briefly expressed by The vanishing of the curvature tensor is the necessary and sufficient condition for the reducibility of g. to a constant tensor, or the criterion of a homaloidal (Euclidean) space. In fact, if g.,(x) is constant throughout S., all Christoffel symbols and RK.43 vanish in x and therefore also in any co-ordinate-system. Con versely, if Kas vanishes, AX" = o for every circuit, parallelism becomes independent of the route of transfer and applicable to any pair of distant places. This however can be shown to be suf ficient for constructing the Euclidean geometry. Analytically.

where is the oriented surface-element (the suffix v indicating its normal). In general, Kp will depend on posi tion and on orientation. In other words, with regard to curvature, may be non-homogeneous as well as anisotropic (e.g., space time within or around matter) ; but if Kp is everywhere isotropic, it is also constant throughout S„. This is Schur's theorem. By (1o) the necessary and sufficient condition for isotropy of Riemannian curvature becomes (A, KA) =K(pcgxp,—g,pg),K) or Ra,a= with constant K.

The tensor (9), being mixed, yields the contracted curvature tensor /?",x,,= which turns out to be symmetrical. Its scalar, R= g"R.,, is the curvature invariant of the manifold. For an isotropic n-fold, R= and R.,= g‘KR/n. A capital use of R., was made by Einstein for constructing the gravitational field-equations, KT., where KCB gravitation con stant, and T., is the tensor of matter, embodying energy, mo mentum, and stress. The covariant derivative of R is con nected with grad R by the n relations =18R/ax,, which were again utilized by Einstein and which follow from the iden tical relations /),(LK, vX) = o. These remarkable identities, discovered by Bianchi, give also a very simple proof of Schur's theorem.