Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 5

Section Metric tensor

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Metric tensor

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We shall limit ourselves to a Euclidean space for which we have defined the scalar product. The analytical expression of the scalar product of two vectors [{\bf x} = x^{i}{\bf e}_{i}] and [{\bf y} = y\hskip 2pt^{j}{\bf e}_{j}] is [{\bf x} \cdot {\bf y} = x^{i}{\bf e}_{i} \cdot y\hskip 2pt^{j}{\bf e}_{j}.]Let us put [{\bf e}_{i} \cdot {\bf e}_{j} = g_{ij}. \eqno(]The nine components [g_{ij}] are called the components of the metric tensor. Its tensor nature will be shown in Section[link]. Owing to the commutativity of the scalar product, we have [g_{ij}= {\bf e}_{i} \cdot {\bf e}_{j} = {\bf e}_{j} \cdot {\bf e}_{i} = g_{ji}.]

The table of the components [g_{ij}] is therefore symmetrical. One of the definition properties of the scalar product is that if [{\bf x} \cdot {\bf y} = 0] for all x, then [{\bf y} = {\bf 0}]. This is translated as[x^{i}y\hskip 2pt^{j}g_{ij}= 0 \quad \forall x^{i}\ \Longrightarrow\ y\hskip 2pt^{j}g_{ij} = 0.]

In order that only the trivial solution [(y\hskip 2pt^{j} = 0)] exists, it is necessary that the determinant constructed from the [g_{ij}]'s is different from zero: [\Delta (g_{ij}) \neq 0.]This important property will be used in Section[link].

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