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center o bass

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A connection on P is a unique separation of the tangent space ##T_uP## into the vertical subspace ##V_u P## and the horizontal subspace ##H_u P## such that

$$(i) T_u P = H_u P \oplus V_u P.\\

(ii) \text{A smooth vector field X on P is separated into smooth vector fields}

\ \ X_H \in H_u P \ \ \text{and} \ \ X_V ∈ V_u P \ \ \text{as} \ \ X = X_H + X_V .\\

(iii) H_{ug} P = R_{g∗} H_u P \ \ \text{for arbitrary} \ \ u \in P \ \ \text{and} \ \ g \in G.$$

First of all; is this a generalization of the usual affine connection? Do the affine connection also determine a direct sum composition of the tangent space on a manifold? Are there other similarities?

Any intuitive and enlightening thoughts on why this is a good definition of a connection?