Tangent vectors, differentials, pushforwards, and pullbacks
Continuing the discussion, we now consider a new way of looking at geometrical spaces. This was started by the sailors who needed maps to guide on their perilous journeys. What people discovered is that they cannot draw a map which was free of distortions: either the distance ratios or the angles (and sometimes both) were not preserved. This is because the Earth is spherical (hence curved) and the map is a flat two dimensional surface. From this mathematicians abstracted the idea of a manifold. A manifold is a geometric shape for which we can locally draw a map. Several maps may be needed to completely cover the space (in the Earth’s case one needs a minimum of two maps), and the collection of maps is naturally called an atlas. An atlas must obey a natural requirement: whenever two maps describe a common area, they have to be compatible.
Now at any point in the manifold we can attach a tangent space. How do we represent vectors in this plane?
Here is the first subtle point. The tangent vectors should not be represented by arrows in a plane as in the misleading picture above because this is an embedded picture of the space in a higher dimensional space. As such it can depend on the embedding. What we seek is an intrinsic definition which makes no reference to any embedding. The basis for the tangent plane is given by a linear independent set of partial derivatives along curves passing though the point where the tangent place in defined. Tangent vectors are partial derivatives along some direction!!!
Next consider a map φ between two manifolds. If point x in M is mapped to point y in N, the map of the two tangent planes at x and y is the differential df. Why? Because a tangent vector is a sum of partial differentials multiplied with some weights, and the differential is the Jacobian which performs a change of coordinates in multi-variable calculus.
When you first learn differential topology it is customary to write all this in components and build an intuition by repetitive elementary problem solving. However, if you do not pursue the subject further enough you risk of not seeing the forest because of the horrible complicated trees. What we are after is an intuition in coordinate free language. In differential geometry this means we need to understand the key ideas of pushworward and pullback.
The pushforward is easy to understand. Think of it as a motion picture: a movie on TV is a map between the manifold of where the movie is recorded to the manifold of your screen. Movement on the director’s set is transferred to movement on your screen. In other words, tangent vectors from the domain manifold define tangent vectors in the range manifold.
The pullback is a bit harder to grasp in an intuitive way. A pullback returns differentials from N to M. Differentials come in many forms. One particular important case is when the manifold N is simply the real number line. In this case the differential is called a 1-form. Think of it as a machine which eats a vector and spits out a number. The book Gravitation by Misner, Thorne, and Wheeler has this explained the best. If a vector is pictorially represented as an arrow, a 1-form is represented as an infinite set of parallel planes. The distance between the parallel planes is inverse proportional with the 1-form magnitude, while the direction perpendicular with the planes gives the 1-form a direction. A scalar product between a vector and a 1-form corresponds to how many parallel planes the vector pierces. Misner, Thorn, and Wheeler call this “how many bongs of the bell”. Of course if the vector is parallel with the planes (orthogonal with the 1-form) there are no bongs and the scalar product is zero.
Now back to pullbacks. Imagine at each point on N a 1-form (a set of parallel planes). Can we construct a 1-form on M? Here is how you do it: take a vector from M, push it forward to N, pierce the 1-form and get the number of piercing. Construct on M a 1-form which when pierced by the original vector gives the same “number of bongs”.
Another way to help visualize pullbacks is by Dirac’s electrons and holes idea. Have a hole in the destination manifold N. We pull back the hole from N to M if we push forward an electron from M to N to fill the hole in N and in the process we created a hole in M.
To summarize: tangent vectors are partial derivatives, a differential is a map between tangent planes, we pushforward vectors and we pullback differentials. Next time we’ll marry algebraic with differential topology and arrive at the wonderful theory of De Rham cohomology.