Since mathematical structures are often defined in reference to an underlying
space, many examples of transport of structure involve spaces and
mappings between them. For example, if
V and
W are
vector spaces with (\cdot,\cdot) being an
inner product on W, such that there is an
isomorphism \phi from
V to
W, then one can define an inner product [\cdot, \cdot] on
V by the following rule: :[v_1, v_2] = (\phi(v_1), \phi(v_2)) Although the equation makes sense even when \phi is not an isomorphism, it only defines an inner product on
V when \phi is, since otherwise it will cause [\cdot,\cdot] to be
degenerate. The idea is that \phi allows one to consider
V and
W as "the same" vector space, and by following this analogy, then one can transport an inner product from one space to the other. A more elaborated example comes from
differential topology, in which the notion of
smooth manifold is involved: if M is such a manifold, and if
X is any
topological space which is
homeomorphic to
M, then one can consider
X as a smooth manifold as well. That is, given a homeomorphism \phi \colon X \to M, one can define coordinate charts on
X by "pulling back" coordinate charts on
M through \phi. Recall that a coordinate chart on M is an
open set U together with an
injective map :c \colon U \to \mathbb{R}^n for some
natural number n; to get such a chart on
X, one uses the following rules: :U' = \phi^{-1}(U) and c' = c \circ \phi. Furthermore, it is required that the charts
cover M (the fact that the transported charts cover
X follows immediately from the fact that \phi is a
bijection). Since
M is a
smooth manifold, if
U and
V, with their maps c \colon U \to \mathbb{R}^n and d \colon V \to \mathbb{R}^n, are two charts on
M, then the composition, the "transition map" :d \circ c^{-1} \colon c(U \cap V) \to \mathbb{R}^n (a self-map of \mathbb{R}^n) is smooth. To verify this for the transported charts on
X, notice that :\phi^{-1}(U) \cap \phi^{-1}(V) = \phi^{-1}(U \cap V), and therefore :c'(U' \cap V') = (c \circ \phi)(\phi^{-1}(U \cap V)) = c(U \cap V), and :d' \circ (c')^{-1} = (d \circ \phi) \circ (c \circ \phi)^{-1} = d \circ (\phi \circ \phi^{-1}) \circ c^{-1} = d \circ c^{-1}. Thus the transition map for U' and V' is the same as that for
U and
V, hence smooth. That is,
X is a smooth manifold via transport of structure. This is a special case of transport of structures in general. The second example also illustrates why "transport of structure" is not always desirable. Namely, one can take
M to be the plane, and
X to be an infinite one-sided cone. By "flattening" the cone, a homeomorphism of
X and
M can be obtained, and therefore the structure of a smooth manifold on
X, but the cone is not "naturally" a smooth manifold. That is, one can consider
X as a subspace of 3-space, in which context it is not smooth at the cone point. A more surprising example is that of
exotic spheres, discovered by
John Milnor, which states that there are exactly 28 smooth manifolds which are
homeomorphic but
not diffeomorphic to S^7, the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a
canonical isomorphism between the two objects. == See also ==