# What's the idea behind the covariant derivative?

I'm learning differential geometry from what I find on the Internet (to eventually find a grasp on General Relativity too). Right now I playing with a sphere.

I have 3 functions ($x$, $y$, $z$) that take 2 parameters (latitude and longitude). I denoted it with $\mathbf{y}(\mathbf{x})$.

Then I defined a path $\mathbf{x}(t)$ on this sphere.

If I differentiate $\mathbf y$ with respect to $t$ once I get a tangent vector.

If I differentiate it again, I usually get a vector that's not tangent (so if I walking on a great circle on my sphere, the second derivative, the acceleration, points towards the center of the sphere).

At this point I thought that an inhabitant on this 2D manifold cannot feel accelerations pointing towards the extra-dimensions, only the projections of it. So I projected these vectors into the tangent space and I got this:

$$\frac{d^2 x^i}{dt^2} + \Gamma^i_{jk} \frac{d x^j}{dt} \frac{dx^k}{dt}$$

For latitude circles it's a vector pointing north (on the northern hemisphere).- It makes sense to me, as latitude circles keeps turning northward. But still not sure what I actually did here.

So is this what I did called the covariant derivation (differentiation as usual then projecting the resulting vector into the tangent space)?

What else can I use covariant derivation for?

Sorry if my question looks dumb (compared to others), but I'm not a math student just a curious person who find it easier to grasp concepts if he see them in action.

• There is a book which is fantastic for intuition in differential geometry and relativity: "Gravitation" by Misner, Thorne, and Wheeler. Commented Mar 21, 2014 at 8:37