Elements of Tensor Calculus by A. Lichnerowicz

Author:A. Lichnerowicz [A. Lichnerowicz]
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2016-02-14T16:00:00+00:00

As in Euclidean geometry the quantities

(where the Γkih are as defined above) are the components of a tensor called the covariant derivative of the vector v. If the vector field is defined by the co variant components υi its absolute differential has the covariant components

and the corresponding components of the covariant derivative are given by

The vectors at two neighbouring points M and M′ are said to be identical if their image vectors in a second-order representation are identical; the absolute differential ∇υi corresponding to the passage from the first vector to the second is then zero.

The formulae which define the absolute differential or the covariant derivative of any tensor may also be extended to Riemannian geometry using arguments similar to those above.

To summarize, the concepts of second-order representation and of the osculating Euclidean metric permit the extension to Riemannian spaces of the Euclidean tensor analysis relating to tensors associated with neighbouring points. This holds for all the differential operators we have studied. It is important to note, however, in the case of vectors (for example), that when the absolute differential in Euclidean geometry is an exact differential satisfying the usual integrability conditions, there is no reason to suppose that this will also be true in Riemannian geometry.

76. Acceleration vector of a moving point in Vn. Geodesies. In §66 we considered the motion of a point in Euclidean space. Let us now consider a moving point M in Vn whose position is a function of a parameter t which we shall interpret as the time. The velocity vector of M has the contravariant components

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