## § 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 $M$ of dimension $n$ is a topological space. So, there is a topological structure $T$ on $M$. There is also an Atlas, which is a family of Charts that satisfy some properties.

#### § Chart

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

#### § Atlas

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

#### § Differentiable map

$f: M \to N$ be a mapping from an $m$ dimensional manifold to an $n$ dimensional manifold. Let $frep = cn \circ f \circ cm^{-1}: \mathbb R^m -> \mathbb R^n$ where $cm: M \to \mathbb R^m$ is a chart for $M$, $cn: N \to \mathbb R^n$ is a chart for $N$. $frep$ is $f$ represented in local coordinates. If $frep$ is smooth for all choices of $cm, cn$, then $f$ is a differentiable map from $M$ to $N$.

#### § Curve:

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

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

A differentiable mapping from $M$ to $R$.

#### § 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