Fred wrote: << . . . to define the concept of orientation at all!
For a smooth manifold there are many equivalent definitions of orientation. For Euclidean space the definitions are especially simple (though more complex than one might expect for such a basic idea"). Consider the set B of all ordered bases (v_1,...,v_n) of R^n as a real vector space. B can be considered as a topological space (and metric space), as a subset of the space (R^n)^n = R^(n^2). As such, it has 2 connected components. Definition: An *orientation* of R^n is the choice of one of these components. ---------------------- Given any fixed ordered basis f = (f_1,...,f_n) of R^n, then each element b of B can be expressed uniquely in terms of f, yielding an nxn matrix M_f(b) of real numbers. Then the two components of B can be distinguished by the sign of the determinant of M_f(b). ---------------------- In the more general case of an arbitrary smooth n-manifold X: Each point x of X has a tangent space T_x that naturally has the vector space structure of R^n. So for each x in X there is a space of ordered bases B_x of T_x. Taking the disjoint union of B_x for all x in X, one gets the set of *all* ordered bases at *all* points of X. This is a topological space in a natural way (e.g., as a result of any reasonable metric structure on it). This is called the "Stiefel variety" of X and denoted V(X). (It's also called the "frame bundle" of X, and can be considered as a fibre bundle over X with fibre = the Lie group GL(n).) As a topological space, V(X) is either connected or not. When it is not connected, it has exactly two connected components. Definition: If V(X) has only one component, it's called nonorientable. Definition: When V(X) has two connected components, X is called *orientable*. Definition: If X is orientable, the choice of one of the components of V(X) is called an *orientation* of X. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele