Relation To Principal and Ehresmann Connections
Let E → M be a vector bundle of rank k and let F(E) be the principal frame bundle of E. Then a (principal) connection on F(E) induces a connection on E. First note that sections of E are in one-to-one correspondence with right-equivariant maps F(E) → Rk. (This can be seen by considering the pullback of E over F(E) → M, which is isomorphic to the trivial bundle F(E) × Rk.) Given a section σ of E let the corresponding equivariant map be ψ(σ). The covariant derivative on E is then given by
where XH is the horizontal lift of X (recall that the horizontal lift is determined by the connection on F(E)).
Conversely, a connection on E determines a connection on F(E), and these two constructions are mutually inverse.
A connection on E is also determined equivalently by a linear Ehresmann connection on E. This provides one method to construct the associated principal connection.
Read more about this topic: Connection (vector Bundle)
Famous quotes containing the words relation to, relation, principal and/or connections:
“The psychoanalysis of individual human beings, however, teaches us with quite special insistence that the god of each of them is formed in the likeness of his father, that his personal relation to God depends on his relation to his father in the flesh and oscillates and changes along with that relation, and that at bottom God is nothing other than an exalted father.”
—Sigmund Freud (1856–1939)
“Only in a house where one has learnt to be lonely does one have this solicitude for things. One’s relation to them, the daily seeing or touching, begins to become love, and to lay one open to pain.”
—Elizabeth Bowen (1899–1973)
“So in accepting the leading of the sentiments, it is not what we believe concerning the immortality of the soul, or the like, but the universal impulse to believe, that is the material circumstance, and is the principal fact in this history of the globe.”
—Ralph Waldo Emerson (1803–1882)
“The connections between and among women are the most feared, the most problematic, and the most potentially transforming force on the planet.”
—Adrienne Rich (b. 1929)