The inner product is the outer product (supposed mixed) contracted once or more. Thus, the inner product of A,, BK is AKBK, an invariant; that of AOC, gives A,,,Bx"= C",, a mixed tensor, and, after a second contraction, A,KB"= C',= C, an in variant. (Unlike AKBK, AKBK is not invariant; it has no tensor character, and likewise for A"B". Thus also A., unlike A",, has no invariant of its own.) Again, the inner product of ALK, BX is a covariant vector, etc. Conversely, if AKBK is a scalar for any covariant vector AK, BK is a contravariant vector. Similarly, if A,KB" is a scalar for any covariant A,K, B" is contravariant; if AL"B.x= Ca is covariant for any contravariant A", then Boo, is covariant. This is an efficient method of establishing the tensor character of a set of n, etc. magnitudes.
In what precedes only such properties of tensors were treated as are independent of any metrical considerations, the space S„ being thus far a non-metrical, amorphous manifold. Let now its metrics be fixed by laying down the line-element, a quadratic differential form with coefficients g.= gu, prescribed functions of the x, g (2) to be considered as invariant and to serve as the measure of the (squared) size or length of the vector dx,. Then, dx, being con travariant, g,. will be covariant. Equivalently we may say that a certain symmetrical tensor g. is being impressed upon S„ as the fundamental or metrical tensor, converting it into a metrical manifold, a Riemannian space. Then g,4x,clx. will be an in variant, the squared size, of the vector dx, which had no invariant of its own. In relation to or with the aid of g,K, all other vec tors and higher tensors will now acquire some new properties. These, and these only, will be their metrical properties. Thus, g.ALAK is the invariant squared size or norm of a vector A', g,. B"= B the scalar of B", and g,xgx,,C‘"xAL that of C"'`µ.
The minors r" of g =lg.', divided by g, form again a symmet rical tensor, the contravariant metrical tensor, to be used along with g,.. Thus g"‘X,X,, is the norm of X,. With every vector A, there is metrically associated a contravariant vector g"A.= A',
the conjugate of AL, similarly, g.BK= B,, the conjugate of B'. The conjugate of the conjugate is the original vector. Conjugate vectors, B, and B', have the same size, the norm of either being expressible by BIB". These properties follow from the important formula g.ces = ga = 6.0, where Ls is Kronecker's symbol, o or i, according as afi' or a =0. This symbol, more appropriately is itself a mixed tensor. Similarly, metrical associates are con structed from higher tensors. Thus, to A. belongs A" = its supplement. The supplement of the supplement is the original tensor. With the covariant A.., is metrically associated the mixed tensor A`,= raA.a and the covariant tensor atx= the reduced of ACK.
The angle 0 between two (copunctal) vectors is defined by the invariant cos 0= rA,BK / AB= A".BK/ AB; A, B being their sizes.
For contravariant vectors, cos° = / AB. The unit vector
dx,/ds determines, locally, a direction in S. The angle between two directions it, is given by cos0=p,±4K. Integrals.—The integral f
. . . dx„, briefly f dx, ex tended over a region of 5„, is transformed into f Jdx', and the determinant g into g' = Pg. Consequently, f gdx is an invariant metrically impressed upon that region, its size or volume (area, if n= 2). This concept is readily extended to any sub-manifold of
characterized by x, as functions of m
If, by a proper choice of the co-ordinate system, all components g,. are reducible to constants, the metrical space 5„ is Euclidean or homoloidal (flat). This is but a very special case of a Rie mannian space. In general such a reduction is not possible, and is non-Euclidean (v. infra).
Differentiation of tensors, aided by metrics, yields an unlimited number of new tensors. The oldest of such is Riemann's set of four-index symbols, of 1861. The simplest metrically differential tensor, however, was discovered in 1869 by Christoffel. This is the covariant derivative of a vector At, where i= dx/ds. They determine, e.g., in the case of space-time, the motion of a free particle in the metrical, and gravitational, field g.. The null-lines are imaginary or real according as (2) is a definite or non-definite form. The former is the case of spaces proper as contemplated by the pure geometer, and the latter that of space-time with one positive and three negative g.. The null-lines of space-time represent light propagation. The definiteness or non-definiteness of the quadratic form and its index of inertia (number of negative gar's) are invariant properties.