## The Tangent Space of a Product

Let and be smooth manifolds, with the -dimensional product manifold. Given points and we want to investigate the tangent space of this product at the point .

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.

Wow. Just stumbled on this awesome blog. I’ve set myself a goal of reading all of your back entries so I can “catch up” to the current exposition. Thanks for the (clearly significant) effort you’ve put into this; it’s a great resource!

Comment by Chris | April 28, 2011 |

[...] obvious example of submersion is a projection from a product manifold. As we’ve seen, the determinant of this projection is always a surjection. In fact, it’s a projection [...]

Pingback by Submersions « The Unapologetic Mathematician | May 2, 2011 |

Agree with Chris. This is a great resource and would like to echo the thanks, John. Although I’m not at a sufficient level to grasp all the technical details, this blath provides a great “benchmark” to enable insight into this portion of the “world”. Much appreciated.

Comment by Jubayer | May 11, 2011 |