- #1
eok20
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I'm teaching myself about connections and came across something I am not completely sure about. The text I am using defines a connection as taking in two vector fields and outputting a vector field. However, later when discussing covariant derivatives along a curve I see this equation:
[tex]
D_t V(t) = \nabla_{\dot{\gamma}(t)} W,
[/tex]
where V is a vector field along the curve [tex]\gamma[/tex], W is an extension field of V, [tex]\nabla[/tex] is the connection and D_t takes in a vector field along [tex]\gamma[/tex] and gives a vector field along [tex]\gamma[/tex].
I understand that the left hand side is the value of the vector field D_t V at time t (a tangent vector at [tex]\gamma(t)[/tex]). However, the right hand side is confusing me since [tex]\dot{\gamma}(t)[/tex] is a tangent vector at [tex]\gamma(t)[/tex] and not a vector field. Since the value of a covariant derivative at a point p depends only on the value at p of the vector field we are differentiating along, is the right hand side the same as [tex](\nabla_X W) (\gamma(t))[/tex] where X is any vector field such that [tex]X(\gamma(t)) = \dot{\gamma}(t)[/tex]?
Thanks.
[tex]
D_t V(t) = \nabla_{\dot{\gamma}(t)} W,
[/tex]
where V is a vector field along the curve [tex]\gamma[/tex], W is an extension field of V, [tex]\nabla[/tex] is the connection and D_t takes in a vector field along [tex]\gamma[/tex] and gives a vector field along [tex]\gamma[/tex].
I understand that the left hand side is the value of the vector field D_t V at time t (a tangent vector at [tex]\gamma(t)[/tex]). However, the right hand side is confusing me since [tex]\dot{\gamma}(t)[/tex] is a tangent vector at [tex]\gamma(t)[/tex] and not a vector field. Since the value of a covariant derivative at a point p depends only on the value at p of the vector field we are differentiating along, is the right hand side the same as [tex](\nabla_X W) (\gamma(t))[/tex] where X is any vector field such that [tex]X(\gamma(t)) = \dot{\gamma}(t)[/tex]?
Thanks.