## Tangent vectors, differentials, pushforwards, and pullbacks

### Intuitive cohomology

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

**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**__misleading__**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__. Why? Because a tangent vector is a sum of partial differentials multiplied with some weights, and the differential is the**differential**df**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.
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