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here is the formal question.

M is a riemannian sub-manifold in N. a,b are vector fields such that for each p[tex]\in[/tex]M, a_{p},b_{p}[tex]\in[/tex] T_{p}M [tex]\subset[/tex] T_{p}N

prove

[tex]\nabla[/tex]^{M}_{b}a = pr([tex]\nabla[/tex]^{N}_{b}a)

where pr is the projection funtion pr:T_{p}N[tex]\rightarrow[/tex]T_{p}M

and [tex]\nabla[/tex]^{N}and [tex]\nabla[/tex]^{M}are the covariant derivative operators (by riemannian connection) in N and M respectively.

I don't really understand why is this not immediate from definitions. the covariant derivative is taking a regular derivative and then projecting onto the tangent bundle. in a manifold the tangent bundle is just the manifold itself so the regular derivate is already the covariant derivative because the projection part is just identity. thus when I will project this vector on T_{p}M ofcourse I wil get the covariant derivative on M because it's the projection onto T_{p[\SUB]M of the regular derivative of vector fields on M. is what I'm asked to prove actually just that the covariant derivative equal to the regular derivative when the vector fields in question belong to a sub manifold?}

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# Projection of the co-derivative = co-derivative of the projection ?

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