§ Handy list of differential geometry definitions

There are way too many objects in diffgeo, all of them subtly connected. Here I catalogue all of the ones I have run across:

§ Manifold

A manifold MM of dimension nn is a topological space. So, there is a topological structure TT on MM. There is also an Atlas, which is a family of Charts that satisfy some properties.

§ Chart

A chart is a pair (OT,cm:O>Rn(O \in T , cm: O -> \mathbb R^n. The OO is an open set of the manifold, and cmcm ("chart for "m") is a continuous mapping from OO to Rn\mathbb R^n under the subspace topology for UU and the standard topology for Rn\mathbb R^n.

§ Atlas

An Atlas is a collection of Charts such that the charts cover the manifold, and the charts are pairwise compatible. That is, A={(Ui,ϕi)}A = \{ (U_i, \phi_i) \}, such that iUi=M\cup{i} U_i = M, and ϕjphii1\phi_j \circ phi_i^{-1} is smooth.

§ Differentiable map

f:MNf: M \to N be a mapping from an mm dimensional manifold to an nn dimensional manifold. Let frep=cnfcm1:Rm>Rnfrep = cn \circ f \circ cm^{-1}: \mathbb R^m -> \mathbb R^n where cm:MRmcm: M \to \mathbb R^m is a chart for MM, cn:NRncn: N \to \mathbb R^n is a chart for NN. frepfrep is ff represented in local coordinates. If frepfrep is smooth for all choices of cm,cncm, cn, then ff is a differentiable map from MM to NN.

§ Curve:

Let II be an open interval of R\mathbb R which includes the point 0. A Curve is a differentiable map C:(a,b)MC: (a, b) \to M where a<0<ba < 0 < b.

§ Function: (I hate this term, I prefer something like Valuation):

A differentiable mapping from MM to RR.

§ Directional derivative of a function f(m): M -> R with respect to a curve c(t): I -> M, denoted as c[f].

Let g(t) = (f . c)(t) :: I -c-> M -f-> R = I -> R. This this is the value dg/dt(t0) = (d (f . c) / dt) (0).

§ Tangent vector at a point p:

On a m dimensional manifold M, a tangent vector at a point p is an equivalence class of curves that have c(0) = p, such that c1(t) ~ c2(t) iff :
  • For a (all) charts (O, ch) such that c1(0) ∈ O, d/dt (ch . c1: R -> R^m) = d/dt (ch . c2: R -> R^m).
That is, they have equal derivatives.

§ Tangent space(TpM):

The set of all tangent vectors at a point p forms a vector space TpM. We prove this by creating a bijection from every curve to a vector R^n. Let (U, ch: U -> R) be a chart around the point p, where p ∈ U ⊆ M. Now, the bijection is defined as:
forward: (I -> M) -> R^n
forward(c) = d/dt (c . ch)

reverse: R^n -> (I -> M)
reverse(v)(t) = ch^-1 (tv)

§ Cotangent space(TpM*): dual space of the tangent space / Space of all linear functions from TpM to R.

  • Associated to every function f, there is a cotangent vector, colorfullycalled df. The definition is df: TpM -> R, df(c: I -> M) = c[f]. That is,given a curve c, we take the directional derivative of the function falong the curve c. We need to prove that this is constant for all vectorsin the equivalence class and blah.

§ Pushforward push(f): TpM -> TpN

Given a curve c: I -> M, the pushforward is the curve f . c : I -> N. This extends to the equivalence classes and provides us a way to move curves in M to curves in N, and thus gives us a mapping from the tangent spaces. This satisfies the identity:
push(f)(v)[g] === v[g . f]

§ Pullback pull(f): TpN* -> TpM*

Given a linear functional wn : TpN -> R, the pullback is defined as wn . push(f) : TpM -> R. This satisfies the identity:
(pull wn)(v) === wn (push v)
(pull (wn : TpN->R): TpM->R) (v : TpM) : R  = (wn: TpN->R) (push (v: TpM): TpN) : R

§ Vector field as derivation

TODO

§ Lie derivation

§ Lie derivation as lie bracket