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An affine connection on $M$ is a differential operator, sending smooth vector fields $X$ and $Y$ to a smooth vector field $∇_X Y$ , which satisfies the some conditions.I would like to know the intuition behind the field $∇_X Y$,esp. what quantity it measures.

user122424
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1 Answers1

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The intuition is that, if you ever compute it in the case that $M = \mathbb{R}^n$, a connection is the usual differentiation along the given vector field $X$.

So, a connection on a manifold $M$ should be some way to "differentiate" a vector field along another one. Or: A way to see what's the variation of a given vector field along the second one.

Thus, it can measure how a vector field varies along the trajectories of the other one (for an exact definition of this, look for affine conections along maps).

For example: If the manifold $M$ admits a Riemannian metric, we must want to "differentiate" the inner product of vector fields along a third one. From this, arises the concept of a metric connection: one of the (many) equivalent forms of stating that a connection is metric is that $\forall X,Y,Z \in \Gamma(TM)$ vector fields, we've that

$$ X\langle Y,Z\rangle = \langle \nabla_X Y, Z \rangle + \langle \nabla_X Z, Y \rangle $$

Where the inner product is the one in the right point of the Manifold. Note that this definition gives even more intuition: the usual derivative on $\mathbb{R}^n$ satisfies the given property, and thus an afine metric connection can also be seen as a way to measure how the angle of two vector fields varies along a third one.

João Ramos
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  • I would understand that $∇_X Y$ is how the $Y$ field changes in the direction of the $X$ field.But $∇_X Y$ contains Christoffel's symbols.Why,and what is the interpretation of their presence? – user122424 Jun 27 '15 at 17:31
  • That's a good question; for example, get for the tangent bundle $TM$ a local orthonormal basis ${X_i}$. Then, according to this base, we have that the Christoffel's symbols are just $ \Gamma_{ij}^k = \langle \nabla_{X_i}X_j , X_k \rangle$. That is, we are measuring how the derivative along another vector field beahves in respect to a third vector field in the basis. (We are noting here that the Christoffel's symbols depend on the local frames we select for the tangent bundle). – João Ramos Jun 27 '15 at 18:07
  • In addition, I really am sorry that I cannot provide a full answer to your question about what quantity should $\nabla_X Y$ measure. These things I've said are just what I understand about the action of an affine connection on a vector bundle $\pi :E \rightarrow M$. – João Ramos Jun 27 '15 at 18:29