For some notation, remember that we have the projections and . Also, if we have a point we get a smooth inclusion mapping defined by . Similarly, given a point we get an inclusion map defined by . These maps satisfy the relations
where the last two are the constant maps with the given values. We can thus use the chain rule to calculate the derivatives of these relations
These are four of the five relations we need to show that decomposes as the direct sum of and . The remaining one states
where is a linear map from to . The real content of the first four relations is effectively that
That is, we know that is a right-inverse of , and we want to know if it’s a left-inverse as well. But this follows since both vector spaces and have dimension . Thus the tangent space of the product decomposes canonically as the direct sum of the tangent spaces of the factors. In terms of our geometric intuition, there are directions we can go “along “, and directions “along “, and any other direction we can go in is a linear combination of one of each.
Note how this dovetails with our discussion of submanifolds. The projection is a smooth map, and every point is a regular value. Its preimage is a submanifold diffeomorphic to . The embedding realizing this diffeomorphism is . The tangent space at a point on the submanifold is mapped by to , and the kernel of this map is exactly the image of the inclusion . The same statements hold with and swapped appropriately, which is what gives us a canonical decomposition in this case.