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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:
<br /> D_t V(t) = \nabla_{\dot{\gamma}(t)} W,<br />
where V is a vector field along the curve \gamma, W is an extension field of V, \nabla is the connection and D_t takes in a vector field along \gamma and gives a vector field along \gamma.
I understand that the left hand side is the value of the vector field D_t V at time t (a tangent vector at \gamma(t)). However, the right hand side is confusing me since \dot{\gamma}(t) is a tangent vector at \gamma(t) 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 (\nabla_X W) (\gamma(t)) where X is any vector field such that X(\gamma(t)) = \dot{\gamma}(t)?
Thanks.
<br /> D_t V(t) = \nabla_{\dot{\gamma}(t)} W,<br />
where V is a vector field along the curve \gamma, W is an extension field of V, \nabla is the connection and D_t takes in a vector field along \gamma and gives a vector field along \gamma.
I understand that the left hand side is the value of the vector field D_t V at time t (a tangent vector at \gamma(t)). However, the right hand side is confusing me since \dot{\gamma}(t) is a tangent vector at \gamma(t) 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 (\nabla_X W) (\gamma(t)) where X is any vector field such that X(\gamma(t)) = \dot{\gamma}(t)?
Thanks.