Given two charts with an overlap, we have a transition mapping between the two charts of the overlap, where the mapping corresponds to a position on the manifold.
If this mapping is differentiable, we have a differentiable manifold.
If transition maps are smooth (\(C^\infty\)) then the manifold is smooth.
Take a topological space: can all subsets in the toplogy be mapped to \(n\) dimensional space? if so, manifold
For this we need openness: a graph for example isn’t open and so isn’t a manifold
We also need the same number of dimensions at each point
Isn’t always the case. eg two circles conneceted by a line is not a manifold. it’s 2d in circles, 1d on line (and 3d at connections)
We have a homeomorphism from each point in the toplogy to an n dimensional coordinate system
We also have homeomorphisms of transformation maps, between different points on the topology
The vector space from the homeomorphism is tangent to the manifold at that point. the set of all tangents forms a tangent space
Interior: \(M\); boundry \(\delta M\) Tangent on a manifold:
The tangent space of manifold \(M\) at point \(p\) is denoted \(TM_p\).
If we have a normal field
\(v=v^ie_i\)
Then we can differentiate wrt a direction \(x\).
\(\dfrac{\delta }{\delta x}v=\dfrac{\delta }{\delta x}v^ie_i\)
\(\dfrac{\delta }{\delta x}v=e_i\dfrac{\delta v^i}{\delta x}\)
Because the basis does not change.
If the basis does change we instead have:
\(\dfrac{\delta }{\delta x}v=\dfrac{\delta }{\delta x}v^ie_i\)
\(\dfrac{\delta }{\delta x}v=e_i\dfrac{\delta v^i}{\delta x}+v^i\dfrac{\delta e_i }{\delta x}\)
General point. basis can vary across manifold
After this basis diff
Christoffel symbols are connections.
Torsion tensor is
\(T_{jk}^i=\Gamma^i_{jk}-\Gamma^i_{kj}\)
If torsion is \(0\), then the connection is symmetric.
We can use as the basis for tangent space:
\(\{\dfrac{(\delta }{\delta x^1})_p,(\dfrac{\delta }{\delta x^2})_p,...\}\)
This means we can write a tangent vector as:
\(u=u^i(\dfrac{\delta }{\delta x^i})_p\)
We can use as the basis for the contangent space:
\(\{dx^1,dx^2,...\}\)
The metric depends on the basis too:
\(g_{ij}(p)=g((\dfrac{\delta }{\delta x^i})_p,(\dfrac{\delta }{\delta x^j})_p)\)
The metric on two tangent vectors is defined on the components.
\(g=g_{ij}(p)u^iv^j\)
Essentially as we move across path, we are changing the basis.
We can look at how basis vector change as we translate
We can define as basis as:
\(e_i=\dfrac{\delta x}{\delta x_i}\)
How to measure transport
If we take a vector and move it around a curved surface and return it to the same point, it may not face the same way
Eg if you’re on equator, move east, north, south to equator, you’ll face diffrent direction
This is true on smaller movements of a curved surface
We can use this to measure curvature of a manifold without coordinates
covariant derivative. how does change in field compare to parallel transport from curent position?
\(\nabla_v (X)=\lim_{t\rightarrow 0}\dfrac{X(p+tv)-X(p)}{t}\)
We have point \(p\). We can compare how field in tangent space varies in direction of \(v\).
we don’t define basis as each point, but rather how basis changes as you move along a curve
If we have a tangent vector at one point of the manifold, we can map it to a tangent vector at a nearby point on the manifold.
We can use chain rule. so we can have coordinate maps where there is no overlap.
We have a vector in a tangent space
We have a curve on the manifold from the start point
As we "roll" the tangent, there is a unique vector in each new tangent, determined by transition map
These are affine transformations
Given two points, what path? what transfomration? if curuved then different paths will given different transformation.
Move and therefore change basis, but components are the same.