Let $X, Y$ be two tangent vector fields on a manifold.
Consider their composition:
$(X \circ Y)(f) = X(Y(f)) = X^i\frac{\partial Y(f)}{\partial u^i} = X^i\frac{\partial Y^j}{\partial u^i}\frac{\partial f}{\partial u^j}+X^iY^j\frac{\partial (\frac{\partial f}{\partial u^j})}{\partial u^i}$
$(X \circ Y) = X^i\frac{\partial Y^j}{\partial u^i}\frac{\partial}{\partial u^j}+X^iY^j\frac{\partial (\frac{\partial}{\partial u^j})}{\partial u^i}$
I found that if the part of the second derivative can be expressed as the elements of tangent space $TM$, then it turns out to be the local expression of "Connection" which I just learned recently
$D_X Y = X^i(\frac{\partial Y^j}{\partial u^i}+Y^k\Gamma^j_{ki})\frac{\partial}{\partial u^j}$
So could I consider the connection as a design that allows us to replace the second derivative in order to get some good properties?
Any advice is appreciated