Pullback of Smooth Functions and Smooth Maps
Let φ:M→ N be a smooth map between (smooth) manifolds M and N, and suppose f:N→R is a smooth function on N. Then the pullback of f by φ is the smooth function φ*f on M defined by (φ*f)(x) = f(φ(x)). Similarly, if f is a smooth function on an open set U in N, then the same formula defines a smooth function on the open set φ-1(U) in M. (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions on N to the direct image by φ of the sheaf of smooth functions on M.)
More generally, if f:N→A is a smooth map from N to any other manifold A, then φ*f(x)=f(φ(x)) is a smooth map from M to A.
Read more about this topic: Pullback (differential Geometry)
Famous quotes containing the words smooth, functions and/or maps:
“Success can make you go one of two ways. It can make you a prima donna, or it can smooth the edges, take away the insecurities, let the nice things come out.”
—Barbara Walters (b. 1931)
“When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
—Edward T. Hall (b. 1914)
“And at least you know
That maps are of time, not place, so far as the army
Happens to be concerned—the reason being,
Is one which need not delay us.”
—Henry Reed (1914–1986)