- #1

cianfa72

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- TL;DR Summary
- About the canonical/natural identification of tangent bundle over an affine space with product bundle

Hi,

as in this thread Newton Galilean spacetime as fiber bundle I'd like to clarify some point about tangent bundle for an Affine space.

As said there, I believe the tangent space ##T_pE## at every point ##p## on the affine space manifold ##E## is canonically/naturally identified with the "translation" vector space ##V## of the affine structure ##(E,V)## (i.e. such identification does not require any arbitrary choice).

$$ T_pE \cong V$$ From the above it should follow that ##\tau(E)##, the tangent bundle over ##E##, is canonically identified with the product bundle ##E\times V##.

Is the above correct ? Thanks.

as in this thread Newton Galilean spacetime as fiber bundle I'd like to clarify some point about tangent bundle for an Affine space.

As said there, I believe the tangent space ##T_pE## at every point ##p## on the affine space manifold ##E## is canonically/naturally identified with the "translation" vector space ##V## of the affine structure ##(E,V)## (i.e. such identification does not require any arbitrary choice).

$$ T_pE \cong V$$ From the above it should follow that ##\tau(E)##, the tangent bundle over ##E##, is canonically identified with the product bundle ##E\times V##.

Is the above correct ? Thanks.