# 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 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

Then should be a tangent vector. Then there’s also the definition involving derivations. Namely for a manifold consider the set of smooth functions ( say for a real manifold). Then as usual for there are derivation at p: satisfying R-linearity and

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 , I get a derivation via

Conversely, given a derivation at p, in local coordinates D looks like

Then set .

Lastly, there is the cotangent space definition. Basically for a point p, consider the ideal of all smooth functions that vanish at p. Then set the tangent space to be the dual of , 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 , it restricts to a linear map . Additionally, from the L(ei)bnitz rule it descends to a linear map . Conversely, given a linear functional it can be checked that

is a derivation.

### Connections

This is will be short because details are tedious. The motivation is this. Given a manifold , I can talk about vector bundles . The cotangent bundle is a specific example, and as it happens its sections are differentials, i.e.

And in this case there’s a natural operator that takes a section of the contangent bundle to a section of , for example in this case, the seciton is mapped to the section

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 .

Now to describe its enough to describe it on a frame for its sections (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:

Thus to describe in fact its enough to describe , now in local coordinate

So really its enough to know the , but recalling

I find that

So in fact to describe the connection its enough to specify these numbers which I think are called the Christoffel Symbols.

## About this entry

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- Published:
- July 10, 2009 / 9:29 pm

- Category:
- alg. geo., wall scribble

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