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.

 

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