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The connection determines whether space is flat or not.

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WWGD

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No, whether the connection is zero depends on the coordinates. For example, in flat space, cartesian coordinates have zero connection, but spherical coordinates have a nonzero connection.

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WWGD

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Apologies, I was referring to the metric in general, the Euclidean metric tensor:

content://media/external/file/11536

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The covariant derivative of a vector field with respect to a vector in flat Euclidean space is just the directional derivative of the vector field along a curve fitting the vector.

A general covariant derivative is analogous to a directional derivative. This analogy can be expressed formally by the way a vector acts on a vector field. Using the mathematical notation ##∇_{X_{p}}Y## for the covariant derivative of the vector field ##Y## with respect to the tangent vector ##X_{p}## at the point ##p##,the formal properties are

1) ##∇_{X_{p}}(aY+bZ)= a∇_{X_{p}}Y+b∇_{X_{p}}Z## for constants ##a## and ##b##

2) ##∇_{aX_{p}+bZ_{p}}Y = a∇_{X_{p}}Y+b∇_{Z_{p}}Y##

3) ##∇_{X_{p}}fY = df(X_{p})Y + f∇_{X_{p}}Y## for an arbitrary function ##f##. This is called the Leibniz rule

Notice that this definition does not depend on a choice of coordinates. Interestingly these is also no use of a metric tensor so curvature is defined even when there are no ideas of length and angle.

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You do, however, need at least a topology on the manifold in the first place.Notice that this definition does not depend on a choice of coordinates. Interestingly these is also no use of a metric tensor so curvature is defined even when there are no ideas of length and angle.

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martinbn

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But the manifold has topology.You do, however, need at least a topology on the manifold in the first place.

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But the original post asked specifically about Christoffel symbols. I don't think you can make sense of those without a choice of coordinates. You certainly can't say, in a coordinate-independent way, that they are zero or nonzero.

The covariant derivative of a vector field with respect to a vector in flat Euclidean space is just the directional derivative of the vector field along a curve fitting the vector.

A general covariant derivative is analogous to a directional derivative. This analogy can be expressed formally by the way a vector acts on a vector field. Using the mathematical notation ##∇_{X_{p}}Y## for the covariant derivative of the vector field ##Y## with respect to the tangent vector ##X_{p}## at the point ##p##,the formal properties are

1) ##∇_{X_{p}}(aY+bZ)= a∇_{X_{p}}Y+b∇_{X_{p}}Z## for constants ##a## and ##b##

2) ##∇_{aX_{p}+bZ_{p}}Y = a∇_{X_{p}}Y+b∇_{Z_{p}}Y##

3) ##∇_{X_{p}}fY = df(X_{p})Y + f∇_{X_{p}}Y## for an arbitrary function ##f##. This is called the Leibniz rule

Notice that this definition does not depend on a choice of coordinates. Interestingly these is also no use of a metric tensor so curvature is defined even when there are no ideas of length and angle.

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Point well taken.But the original post asked specifically about Christoffel symbols. I don't think you can make sense of those without a choice of coordinates. You certainly can't say, in a coordinate-independent way, that they are zero or nonzero.

I had trouble parsing the language, particularly the words "does the connection always equal zero" when parallel translating. It seemed from the language that there might some confusion about what a connection is. From other posts I thought it important to point out that connections are not coordinate dependent but that their expressions in different coordinate systems are different. This could have been a disconnect in my understanding of the language for instance when you wrote in post #5 that the connection is zero in some coordinate systems but not in others.

Also @PeroK 's point in post #2 that the connection determines the curvature states that the correct test of flatness is that the curvature tensor is identically zero. This is a coordinate free statement. I thought it important to underscore this by describing the affine connection in axiomatic terms rather than in terms of local coordinates.

The OP asked about more general connections not using Christoffel symbols. One way to think about this might be to look at affine connections on other vector bundles than the tangent bundle. In those cases there are no Christoffel symbols. Still curvature is well defined and a connection is flat if the curvature tensor is identically zero. A vector field in the bundle is parallel along a curve on the manifold its its covariant derivative is zero. All of the formal structure is the same.

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You need a smooth differentiable structure not just a topology.You do, however, need at least a topology on the manifold in the first place.

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