Abstract
The word isometry comes from the Greek word \({\mathop {\eta }\limits ^{\prime }}\) \(\iota \sigma o \mu \varepsilon \tau \varrho \acute{\iota }\alpha \) which means the equality of measures. The origin of the modern concept of isometry is rooted in that of congruence of geometrical figures that Euclid never introduced explicitly, yet implicitly assumed when he proceeded to identify those triangles that can be superimposed one onto the other.
The art of doing mathematics consists in finding that special case which contains all the germs of generality.
David Hilbert
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Notes
- 1.
Clarification for mathematicians: in the physical literature linear representation of a symmetry corresponds to the case where the fundamental fields spanning the theory transform in a linear representation of the considered Lie group G. The Lagrangian defining the considered theory is supposed to be invariant with respect to such transformations. On the other hand the wording non-linear representation is universally used when the fundamental fields of the theory are identified with the coordinates of a Riemannian manifold \(\mathscr {M}\) on which the Lie group G acts as a group of isometries. Indeed in order to be a symmetry of the theory, the action of the group G must leave the lagrangian invariant and this implies the existence of an invariant metric g on \(\mathscr {M}\). The metric g appears in the kinetic term of the fields.
- 2.
Clarification for mathematicians: in the physical literature it is universally utilized the following jargon which turns out to be very clear to readers with an education as physicists. A Lie group element \(g\in \mathbb {G}\) is named infinitesimally close to the identity when its Taylor series expansion in terms of a parameter \(\varepsilon \) that parameterizes a one-dimensional subgroup \(\mathscr {G} \subset G\) to which g belongs is truncated to the first order term: \(g \, = \, e+ \varepsilon \, \mathbf {g} \, + \, \mathscr {O}(\varepsilon ^2)\). Clearly the coefficient \(\mathbf {g}\) of the first order term is an element of the Lie algebra \(\mathbb {G}\) of \(\mathrm {G}\). Applying this jargon to the case of the group of diffeomorphisms, by means of a diffeomorphism infinitesimally close to the identity we define a vector field, the Lie algebra of the diffeomorphism group being the Lie algebra of vector fields. In the case the considered infinitesimally close to identity diffeomorphism is an isometry, the corresponding vector field is named a Killing vector field.
- 3.
We assume that G is semi-simple so that the Cartan-Killing metric is non degenerate.
- 4.
The proof is also summarized in Appendix B of [5].
- 5.
A solvable Lie algebra s is completely solvable if the adjoint operation \(\mathrm {ad}_X\) for all generators \(X \in s\) has only real eigenvalues. The nomenclature of the Lie algebra is carried over to the corresponding Lie group in general in this chapter.
- 6.
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Fré, P.G. (2018). Isometries and the Geometry of Coset Manifolds. In: Advances in Geometry and Lie Algebras from Supergravity. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-74491-9_2
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