The problem of defining the derivative of a vector field in the direction
For a vector field near , in the direction, try to calculate the derivative in coordinates
However, the difference operation is not linearly compatible with coordinate changes of general diffeomorphisms.
But in a metric-manifold, there are special coordinates โ geodesic coordinates. The transformation method for different geodesic coordinates at is , which is linear.
[geodesic-derivative] Geodesic derivative alias [Levi-Civita-derivative] Levi-Civita derivative :=
In geodesic coordinates at point , the derivative at point ,
It is also possible to take the derivative of a tensor field . According to the scalar multiplication associated with the tensor structure, calculations can be performed using the product-rule Example
Prop . Proof In geodesic coordinates
Prop or
Prop The covariant derivative is compatible with the metric-dual e.g. since
Other coordinates may be needed to compute geodesic coordinates, and thus other coordinates may also be needed to express the geodesic derivative.
[geodesic-derivative-in-general-coordinate]
Compute geodesic coordinates using general coordinates , then in coordinates , the geodesic derivative is
Using connection transformation
==>
Use . Substitute into the calculation of
The tangent space linearly transforms to by , but keeps in coordinate , but keep in coordinate
Or written as, in general coordinates, geodesic derivative
For coordinate-frame
Is there a more intuitive explanation, rather than directly using the transformation of connection?
If we only consider linear compatibility, then there are many linear connections, and the one that coincides with the geodesic-derivative is the metric-connection
[geodesic-derivative-of-co-vector] Prop For co-vector field
Proof
Question Similar to the case of vector fields. Use the transformation and the product-rule
For co-vector coordinate-frame
[parallel-transport-metric-connection]
Parallel transport as "zero rate of change along the curve" or where
is an ODE
According to calculation (?), covariant derivative can be recovered from parallel transport + difference quotient
[orthonormal-frame]
Parallel transport of metric-connection preserves the metric
Can be used to construct orthonormal frames
It can be proven that a manifold metric uniquely corresponds to an principal-bundle structure on the manifold
But are there more specific and operational calculation results? Regarding calculating orthonormal frames using parallel transport in geodesic coordinates
A canonical orthonormal frame may be used for simplified some of calculations of spinors on curved manifolds, e.g. Pauli-matrix