Intrinsic Tangent Space

The tangent space at a point  in an abstract manifold  can be described without the use of embeddings or coordinate charts. The elements of the tangent space are called tangent vectors, and the collection of tangent spaces forms the tangent bundle.

One description is to put an equivalence relation on smooth paths through the point . More precisely, consider all smooth maps  where  and . We say that two maps  and  are equivalent if they agree to first order. That is, in any coordinate chart around , . If they are similar in one chart then they are similar in any other chart, by the chain rule. The notion of agreeing to first order depends on coordinate charts, but this cannot be completely eliminated since that is how manifolds are defined.

Another way is to first define a vector field as a derivation of the ring of smooth functions . Then a tangent vector at a point  is an equivalence class of vector fields which agree at . That is,  if  for every smooth function . Of course, the tangent space at  is the vector space of tangent vectors at . The only drawback to this version is that a coordinate chart is required to show that the tangent space is an -dimensional vector space.