# Calcolo geometrico, secondo l'Ausdehnungslehre di H. by Peano G. By Peano G.

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52) in mind, in Euclidean space described by Cartesian coordinates, the orthonormal basis g | x , . . , -^ naturally induces orthonormality for the corresponding basis of the dual space gij = g*(dx\dxj) = Sij . 56) In Minkowski space described by coordinates (x° = c ^ x ^ x ^ x 3 ) , related orthonormal basis (gfs, gfr, gjy, gfr) and metric rj = diag(l, —1, —1, —1),

11) of ® dx J 1 • • • <8> d x J r (CJ1 , . . •<•••<. , dxJT (u1,. ,uq,vi,.. 13) 42 Geometrical Properties of Vectors and Covectors So, any tensor T of type (q,r) can be written as T = T j : ; ; 9 — J - ® . . ® _ _ ® tfaJi ... 11). 9) spans T p ( "' r) (M). 9) are linearly independent. 14) is taken to mean that the result of evaluating it on any set (u1,... ,uq, v\,... ,vr) is zero. , dxk", ^ , . . , •£l7) should produce T h"A? ITT ® • • • ® ITT- ® dxh 3l-3r Qxn ®• • • Qxlq d dx'1 V ' <9x* = 0.

The metric could equally have been associated to the collection of tangent spaces TpM, considered as identical realizations of the vector space V. This tangent space metric exhibits the constant nature of the metric over the whole of the affine space. This constancy has to be abandoned when generalizing metric to arbitrary manifolds [CP86]. A manifold M is said to possess a metric structure if a non-degenerate, symmetric tensor gp of covariant order two, is assigned on each tangent space TpM, varying smoothly with p.