Does anyone know the exact difference between a tangent bundle and a vector field? Mathworld says: A vector field is a tangent bundle section of its tangent bundle. The problem is that using the ordinary definitions, it appears that a vector field is equivalent to a tangent bundle, not just a section. Can someone clarify this?
Hi Mannyfold, welcome to SciForums, A vector field in physics is just a vector-valued function over space. I don't know what a tangent bundle is. Are you trying to describe some surface to which a vector field is tangent at all points? -Dale
Hi mannyfold, Let me try to clarify things for you. The tangent space to manifold M at a point p, T<SUB>p</SUB>M, is the place where all the tangent vectors at p live. The tangent bundle is then the set of all ordered pairs (p,v) where p is a point in M and v is a point in T<SUB>p</SUB>M. In other words, each element of the tangent bundle is a pair consisting of a point p and a tangent vector from T<SUB>p</SUB>M. A vector field is a function f:M -> TM which takes a point p in M and gives you an ordered pair (p,v(p)) in TM. Different vector fields correspond to different ways to choose a vector in the tangent space at p. So while a vector field is a function whose range is contained in TM, it is not the tangent bundle itself. For a general fiber bundle F over M, a section of that bundle is a map s:M -> F which satisfies the addition requirement that s(p) "lies over p," or in technical terms, π(s(p)) = p, where π is the projection associated with the fiber bundle. You can also say that a section maps a point p to an element of the fiber π<SUP>-1</SUP>(p). How ever you phrase it, this is exactly what a vector field does. Hope this helps.
