The problem: you cannot subtract arrows at different points
By now the ground under your feet is a manifold, and at each point P it carries a tangent space — the flat vector space of all the directions you could set off in from P. The previous guide handed you a Riemannian metric, an inner product living in each tangent space, so you can already measure the length of any one arrow and the angle between any two arrows at the same point. That is a lot. But it leaves a surprisingly basic operation broken: you cannot compare a vector at one point to a vector at a different point.
In flat space this is so easy we never notice the gift. Two arrows sitting at different places in a plane? Just slide one over until its tail meets the other's tail and compare — the act of 'sliding without turning' is free, because every tangent plane of the flat plane is secretly the same plane. On a curved manifold that freebie evaporates. The tangent space at the North Pole of a sphere and the tangent space at the equator are genuinely different two-dimensional planes, tilted differently in space. There is no canonical, built-in way to say which equator-arrow 'is the same as' a given pole-arrow. Subtraction, and therefore the derivative of a vector field, has nothing to stand on.
The connection: a rule for differentiating a vector field
The fix is to declare the missing comparison, by adding one new piece of structure called a connection. The honest way to phrase what it does is through differentiation. Suppose you have a vector field Y — an arrow chosen smoothly at every point — and you want to know how Y changes as you move in some direction v. The naive answer is to take Y at a nearby point, subtract Y here, divide by the small distance, and take a limit. But we just saw that subtraction across points is meaningless. The connection supplies exactly the missing ingredient: a way to carry Y from the nearby point back to here so the subtraction makes sense. The result is written nabla_v Y, the covariant derivative of Y in the direction v.
Read nabla_v Y as 'the honest rate of change of the arrow field Y as you step in direction v, with the spurious turning that comes from curved coordinates already subtracted off.' It is itself a tangent vector at the point, pointing the way Y is genuinely twisting and stretching. A connection is precisely a recipe that assigns such a covariant derivative to every pair (direction v, vector field Y), obeying the natural rules of a derivative: it is linear in the direction v, linear in Y, and satisfies the product rule when you scale Y by a function — nabla_v (f Y) = (v applied to f) Y + f nabla_v Y. Those rules are non-negotiable; they are what make nabla deserve the name derivative.
Christoffel symbols: the connection in coordinates
To compute anything we drop into a coordinate patch, with coordinate directions partial_1, partial_2, ... , partial_n — the basis tangent vectors along each coordinate curve, exactly like the x_u and x_v of the surfaces rung, now in n dimensions. The connection is pinned down completely once we know how it differentiates each basis field along each basis direction. That single bundle of numbers is the famous Christoffel symbols, written Gamma^k_ij. The defining equation is short: nabla of partial_j in the direction partial_i is the combination Gamma^k_ij partial_k, summed over k.
Written out, nabla_(partial_i) partial_j = sum over k of Gamma^k_ij partial_k — so the Gamma^k_ij are simply the components, in the partial_k basis, of how the j-th coordinate arrow changes as you step along the i-th direction. Two warnings, told straight. First, the Christoffel symbols are not the components of a tensor — they transform with an ugly extra term when you change coordinates, precisely because their job is to cancel the spurious turning that coordinates introduce. On flat space in straight Cartesian coordinates every Gamma^k_ij is zero; switch to polar coordinates on that same flat plane and they become nonzero, even though nothing curved. So a nonzero Christoffel symbol does not mean the space is curved — it can simply mean your coordinate grid is bent. Second, a bare manifold admits infinitely many different connections; there is no single 'right' one until we ask for more.
Here is where the metric earns its keep. Among all those candidate connections, exactly one is singled out by two reasonable demands: that it be torsion-free (the symmetry Gamma^k_ij = Gamma^k_ji, so there is no built-in twisting) and that it be metric-compatible (it preserves the metric, so carrying arrows along never secretly distorts lengths or angles). The Fundamental Theorem of Riemannian Geometry says these two conditions pin down a unique connection, the Levi-Civita connection, and they even give a closed formula for its Christoffel symbols purely in terms of the metric g and its first derivatives. The metric you already have therefore determines its own notion of differentiation — no extra choice required. From here on, the connection means this one.
Parallel transport: carrying an arrow without turning it
Now that we can differentiate, we can finally define 'sliding an arrow without turning it.' Take a path through the manifold and a vector riding along it. We say the vector is parallel-transported if its covariant derivative along the path is zero at every instant — nabla of the vector, in the direction of the path's own velocity, equals 0. In words: relative to the connection, the arrow is not changing at all; whatever turning you see is only the manifold curving beneath it, never any intrinsic spin of the arrow. This is the curved-space replacement for 'keep it pointing the same way' — and because the connection is metric-compatible, parallel transport keeps the arrow's length fixed and preserves angles between two arrows carried together.
Here comes the punchline that makes the whole subject worth the trouble, and it is best felt with a globe. Stand at the North Pole holding an arrow that points toward, say, the meridian through London. Walk straight down that meridian to the equator, keeping the arrow parallel the whole way — it still points along your meridian, now lying flat and pointing south. Now walk a quarter of the way around the equator, parallel-transporting all the way; the arrow keeps pointing south the whole stroll. Finally walk straight back up to the North Pole. You arrive holding an arrow that has turned 90 degrees from the one you set out with — even though you never once twisted it by hand.
Geodesics: the straightest possible paths
With parallel transport in hand we can finally say what 'straight' means on a curved manifold, and the definition is beautifully self-contained. A straight line in flat space is the path that never turns — its velocity vector keeps pointing the same way the whole journey. Translate 'keeps pointing the same way' into the language of the connection and you get: a curve is a geodesic if its own velocity vector is parallel-transported along the curve. Equivalently, the covariant derivative of the velocity along the path is zero — the path accelerates only in the unavoidable way the manifold forces, with no extra sideways steering of its own. A geodesic is a path that is, by its own internal accounting, going perfectly straight.
Unfolded in coordinates, the parallel-velocity condition becomes the geodesic equation, a system of second-order differential equations: the coordinate acceleration of the curve plus a Christoffel-symbol term built from its velocity equals zero. The shape is worth seeing once, because it makes the role of the Gamma's vivid: in flat Cartesian coordinates every Gamma vanishes and the equation collapses to 'acceleration = 0', i.e. ordinary straight lines at constant speed — exactly what you expect. The Christoffel term is the precise correction that bends the notion of straight to fit a curved space or a curved coordinate grid.
the geodesic equation (curve x^k(t), dot = d/dt)
ddot x^k + sum over i,j of Gamma^k_ij dot x^i dot x^j = 0
flat Cartesian coords: all Gamma = 0 => ddot x^k = 0
=> straight lines at constant speed
the exponential map exp_P(v) : fire a geodesic from P with
initial velocity v, follow it for one unit of time, land here.Two honest cautions before we celebrate. A geodesic is the straightest path, captured by zero covariant acceleration; on a Riemannian manifold it is also locally the shortest path between nearby points, but only locally. A great circle on a sphere is a geodesic, yet the long way around it from London to Tokyo is still a geodesic while being far from the shortest route — straightest and globally-shortest are not the same claim, and conflating them is a common slip. Second, geodesics can stop short: a manifold with a puncture or an edge may have geodesics that run off it in finite time. A manifold where every geodesic extends forever is called geodesically complete, a property we must check, never assume.
Putting it to work, and where it leads
Step back and see the chain you have built. The metric gives you lengths and angles at each point; it manufactures its own Levi-Civita connection, encoded in the Christoffel symbols; the connection tells you how to parallel-transport an arrow along any path; and a path that parallel-transports its own velocity is a geodesic, the curved world's straight line. Every link is forged from the metric alone, intrinsically, with no appeal to any space the manifold might sit inside. The exponential map seals the picture: stand at a point P, pick a tangent direction v, fire off the geodesic in that direction and ride it for one unit of time, and where you land is exp_P(v). It converts arrows in the flat tangent space into actual points on the curved manifold — straight-line travel made literal.
- Start from the metric g in your coordinates; from g and its first derivatives compute the Christoffel symbols Gamma^k_ij of the Levi-Civita connection.
- To carry an arrow along a path, solve the parallel-transport equation (covariant derivative along the path = 0); the arrow's length and angles are automatically preserved.
- To find a geodesic, solve the geodesic equation ddot x^k + Gamma^k_ij dot x^i dot x^j = 0 with a starting point and an initial velocity.
- Transport an arrow once around a closed loop and measure the angle it returns rotated by — a nonzero angle is curvature, and the doorway into the next guide.
And this is exactly where the road to general relativity opens, though we only point down it here. Einstein's insight was that gravity is not a force pulling objects off straight lines, but the curvature of spacetime itself — and a freely falling planet or photon simply follows a geodesic, going as straight as the curved geometry allows. The Christoffel symbols you just met reappear there as the gravitational field, and the geodesic equation becomes the law of free fall. We have not earned that claim yet: it needs the curvature tensor, the topic of the next guide, and the spacetime metric, the topic of the last. But every gear in that machine — connection, parallel transport, geodesic — you have now built honestly, from the inside out.