Page 80 -

action λ. The orbit of a point x ∈ X is the subset Levi-Civita connection The symmetric
o ={x ∈ X : ∃g ∈ G, λ(g, x) = x }. The (linear) connection N α on a (pseudo)-

x βµ
action is transitive if o = X (for any x ∈ X). Riemannian manifold (M, g) uniquely defined
x
The action is free if it has no fixed points, by the requirement that the metric g is parallel,
i.e., if there exists an element x ∈ X such that i.e., ∇g = 0. Equivalently, it is the only
α
λ(g, x) = x, then one has g = e. Example: connection N βµ such that
3
Rotations in R are not free and not transitive; (i.) it is symmetric, i.e., torsionless, i.e.,
translations are free and transitive. N βµ = N ;
α
α
µβ
If X has a further structure, one usually
(ii.) it is compatible with the (pseudo)-metric
requires λ to preserve the structure. For exam-
structure, i.e.,
ple, if X is a topological space, λ are required
g
"
"
to be continuous for any g ∈ G. ∂ g µν − N µλ g − N νλ g µ" = 0.
λ
"ν
If X = V is a vector space, λ are required to
g The coefficients of the Levi-Civita connection
be linear for any g ∈ G and in this case the left α * +
α
are usually denoted by N = and can be
action is also called a representation of G on V . βµ βµ g
expressed as a function of the (pseudo)-metric
If X = M is a manifold, λ are required to be
g
diffeomorphisms for any g ∈ G. tensor and its first derivatives; they are called
If λ is a left action, then one can define a right Christoffel symbols.
−1
action by setting ρ(x, g) = λ(g ,x).
Lewis acid A molecular entity (and the cor-
responding chemical species) that is an electron-
left invariance The property that an object
pair acceptor and, therefore, able to react with a
on a manifold M is invariant with respect to a left
Lewis base to form a Lewis adduct, by sharing
action of a group on M. For example, a vector
the electron pair furnished by the Lewis base.
field X ∈ X(M) is left invariant with respect to
the left action λ : M → M if and only if:
g Lewis adduct The adduct formed between
a Lewis acid and a Lewis base.
T λ X(x) = X(g · x).
x g
Lewis base A molecular entity (and the cor-
left translations (on a group G) The left
responding chemical species) able to provide a
action of G onto itself defined by λ(g, h) =
pair of electrons and thus capable of coordina-
λ (h) = g · h. Notice that, if G is a Lie group,
g
λ g ∈ Diff(G) is a diffeomorphism but not a tion to a Lewis acid, thereby producing a Lewis
homomorphism of the group structure. See also adduct.
right translations and adjoint representations. Lie algebra A vector space L endowed with
an operation [ , ]: L × L → L having the
Legendre transformation Given a Lagran-
gian L : TQ → R, the Legendre transformation following properties
FL : TQ → T Q is defined in local coordinates (i.) [ , ] is bilinear;
∗
i
i
(q , ˙q ) of TQ by (ii.) [ , ] is antisymmetric (or skew-
symmetric), i.e., [A, B] =−[B, A];
∂L
i i i
FL(q , ˙q ) = (q ,p ), where p = . (iii.) [ , ] satisfies Jacoby identity (i.e.,
i
i
∂ ˙q i
[[A, B],C] + [[B, C],A] + [[C, A],B] = 0).
This gives an equivalence between the Euler-
The operation [ , ] is called the Lie bracket or
Lagrange equations and the Hamilton equations commutator. Notice that usually Lie algebras are
of motion.
not associative. See Lie group for the definition
of a Lie algebra of the Lie group G.
length In the context of graph theory, the
Lie derivative Let α be an exterior k-form
length of a path or cycle is the number of edges
and X a vector field with flow ϕ . The Lie deriva-
t
in the particular subgraph.
tive of α along X is given by
Comment: Lengths are an instance of the more
1
general class of measures (such as string length, L α = lim [(ϕ α) − α].
∗
absolute value of a number, etc.). X t→0 t t
© 2003 by CRC Press LLC