# Tangent Space, Derivations, and Connections

There’s lots of different ways to think about tangent space.  From alg. geo. point the tangent space of a variety at a point is a vector space of dimension $\ge$ the dimension of the variety.  Equality holding is what gives smoothness.  The manifold perspective says to take any path passing through your point, in local coordinates this becomes

$\gamma(t) = (\gamma_1(t) ,..., \gamma_n(t))$

Then $\gamma'(t)$ should be a tangent vector.  Then there’s also the definition involving derivations.  Namely for a manifold $M$ consider the set of smooth functions $C = C^\infty( M, \mathbb{R})$ ( say for a real manifold).  Then as usual for $p \in M$ there are derivation at p: $D \colon C \to \mathbb{R}$ satisfying R-linearity and

$D(fg) = f(p)D(g) + D(f)g(p)$

Then define the tangent space to be the set of all derivations.  The relation between the two definition are as follows.  Given a tangent vector of the form $\gamma'(t)$, I get a derivation via

$D(f) = (f \circ \gamma)'(0)$

Conversely, given a derivation $D$ at p, in local coordinates $x_1, ..., x_r$ D looks like

$\sum_i a_i \frac{\partial}{\partial x_i}$

Then set $\gamma'(t) = (a_1, ..., a_r)$.

Lastly, there is the cotangent space definition.  Basically for a point p, consider the ideal $I = I_p \subset C$ of all smooth functions that vanish at p.  Then set the tangent space to be the dual of $I/I^2$, which is the cotangent space (this is like looking at linear behavior).

Here’s the equivalence between cotangent space and derivation definitions.  Given a derivation $D$, it restricts to a linear map $D \colon I \to \mathbb{R}$.  Additionally, from the L(ei)bnitz rule it descends to a linear map $D \colon I/I^2 \to \mathbb{R}$.  Conversely, given a linear functional $m \colon I/I^2 \to \mathbb{R}$ it can be checked that

$D(f) := m( f - f(x) ) = m( f - f(x) + I^2)$

is a derivation.

### Connections

This is will be short because details are tedious.  The motivation is this.  Given a manifold $M$, I can talk about vector bundles $E \to M$.  The cotangent bundle is a specific example, and as it happens its sections $s \colon M \to \mbox{CotBun}$ are differentials, i.e.

$s = \sum_i f_i(x_1, ..., x_r) dx_i$

And in this case there’s a natural $d$ operator that takes a section of the contangent bundle to a section of $\mbox{CotBun}\otimes \mbox{CotBun}$, for example in this case, the seciton $s$ is mapped to the section

$d(s) = \sum_i \sum_j \frac{df}{dx_j} dx_i \wedge dx_j$

So I differentiated a section of the cotangent bundle.  The whole point of connections is to be able to do this for other vector bundles.  The right way to generalize is to look for a map $\Delta \colon E \to E \otimes \mbox{CotBun}$.

Now to describe $\Delta$ its enough to describe it on a frame for its sections $\mu_i \colon M \to E$ (by this I mean any section is some linear combination of these guys).  Then I can compute for a general section by forcing the L(ie)bniz rule:

$\Delta(\sum_i f_i(x_1, ..., x_r) \mu_i)$

$= \sum_i [f_i (x_1, ..., x_r) \Delta(\mu_i) + d(f_i(x_1, ..., x_r) \mu_i]$

Thus to describe $\Delta$ in fact its enough to describe $\Delta(\mu_i)$, now in local coordinate

$\Delta(\mu_i) = \sum_{j,k} a_{j,k} m_j \otimes dx_k$

So really its enough to know the $a_{j,k}$, but recalling

$dx_k (\partial /\partial x_l) = \delta_{kl}$

I find that

$\sum_j a_{j,k}\mu_j = \Delta(\mu_i)(\partial /\partial x_k) := \sum_j \Gamma_{i,k}^j \mu_j$

So in fact to describe the connection its enough to specify these numbers $\Gamma_{i,k}^j$ which I think are called the Christoffel Symbols.