I wrote (for a smooth manifold X): << As a topological space, [the frame bundle] 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.

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Correction: The first definition should read: << Definition: If V(X) has only one component, X is called *nonorientable*.

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--Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele