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I was hoping to gain more intuition about the similarities and differences between the covariant derivative (of any connection, not necessarily the Levi Civita one if it exists) on real and complex manifolds

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This question is outlandishly vague. Specifically, it is almost as vague as asking for intuition on the similarities and differences between vectors in a given vector space. –  J. Martel Dec 17 '13 at 20:13
To be more specific, I was hoping to learn which commonly used properties of the real covariant derivative do not hold (in analogue) for the complex covariant derivative. –  Benjamin Dec 18 '13 at 2:15
As far as I know, a covariant derivative on a complex manifold (whatever you mean by that) is a special case of a covariant derivative on a real manifold. Do you have an example where this is not the case? –  Deane Yang Dec 18 '13 at 14:52
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2 Answers

I have always found the motivation for connections as ``allowing us to differentiate vector fields or tensors by vector fields" outrageous -- but that's my own taste. Connections become appealing to me when I remember Kobayashi-Nomizu's definition of connections on principal fibre bundles. That is, we have a principal fibre bundle $P$ over some manifold $M$ with structure group $G$. For every point $p \in P$ the tangent space $T_pP$ has a canonical direction defined by the orbit of $G$, i.e. those vectors tangent to the orbit through $p$. A connection $\nabla$ then becomes a smooth family of subspaces $Q_p, p\in P$ such that (i) the tangent space at $p$ splits into a direct sum of the tangent space to the orbit $T_p G.p$ and $Q_p$ and (ii) for every $g \in G$, the subspace $Q_{gp}$ coincides with the image of $Q_p$ under the differential $dL_g$ of the left translation by $g$. More briefly, the tangent bundle $TP$ has a canonical subbundle defined by the orbits of $G$, whereas there is no canonical choice of complimentary subbundle. A connection is such a smooth complimentary subbundle.

All these basic facts are in Chapter II (Theory of Connections) in Kobayashi-Nomizu I.

However, I have never been able to understand exactly how torsion or curvature or symmetry translates into this setting.

A second useful reference for the abstract development of connections is Lawson's "Minimal submanifolds in complex geometry". This might be up your alley if you want to compare connections in the complex setting.

On MO, I think you only get as good as you give. I initially downvoted your question because, in all fairness, it deserves to be closed and then resurrected.

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I haven't looked at Kobayashi-Nomizu in decades, but I'd be surprised if it doesn't define curvature using the definition of a connection you cite. In any case, it's easy to guess what it is, since a connection is flat if and only if the distribution is integrable. Probably the easiest way to verify this is by differentiating the Lie algebra-valued connection 1-form. Torsion is defined only if the principal fiber bundle is a subbundle of tangent frame bundle. Not sure how that translates. And what do you mean by symmetry? That's usually a synonym for torsion-free. –  Deane Yang Dec 18 '13 at 14:58
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Let $X$ be a smooth manifold and $\pi:V\rightarrow X$ a real vector bundle. Let $\mathcal{O}_X$, $\Gamma$, and $\Omega_X^1$ denote the sheaves of functions on $X$, smooth sections of $V$, and $1$-forms on $X$, respectively. A connection on $V$ is a morphism $$\nabla:\Gamma\rightarrow\Gamma\otimes_{\mathcal{O}_X}\Omega_X^1$$ of sheaves of $\mathbb{R}$-vector spaces on $X$ that satisfies the Leibniz rule.

Essentially, a connection is just a mechanism for differentiating (local and global) sections of $V$. (It generalizes the exterior derivative of smooth functions on $X$, since the exterior derivative is an example of a connection on the trivial rank-one bundle on $X$.) The covariant derivative comes from a connection on the tangent bundle and allows you to differentiate its sections (vector fields).

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