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 whole point of Camp is to dethrone the serious. Camp is playful, anti-serious. More precisely, Camp involves a new, more complex relation to “the serious.” One can be serious about the frivolous, frivolous about the serious.”
—Susan Sontag (b. 1933)
“Whoever has a keen eye for profits, is blind in relation to his craft.”
—Sophocles (497–406/5 B.C.)
“The principal thing children are taught by hearing these lullabies is respect. They are taught to respect certain things in life and certain people. By giving respect, they hope to gain self-respect and through self-respect, they gain the respect of others. Self-respect is one of the qualities my people stress and try to nurture, and one of the controls an Indian has as he grows up. Once you lose your self-respect, you just go down.”
—Henry Old Coyote (20th century)
“The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.”
—C.G. (Carl Gustav)