What a metric actually is: an inner product in every tangent space
From the bundles rung and Volume I you already have a bare smooth manifold M: a space that looks locally like R^n, with a tangent space T_p M of velocity vectors at each point p. But a bare manifold cannot tell you the length of a curve or the angle between two directions — those words are meaningless until you add structure. A Riemannian metric g supplies exactly that missing structure, and it does so in the leanest possible way: at each point p it is an inner product g_p on the single vector space T_p M. Nothing global, nothing exotic — just a positive-definite symmetric bilinear form on each tangent space, one per point.
The only extra demand is smoothness: as p moves, g_p must vary smoothly, never jumping. In a chart with coordinates x^1, ..., x^n this is wholly concrete. The coordinate vector fields d/dx^i give a basis of each T_p M, and the metric is pinned down by the n-by-n matrix of numbers g_ij(p) = g_p(d/dx^i, d/dx^j), one smooth function per entry. People write the whole thing as the line element ds^2 = g_ij dx^i dx^j (Einstein summation: repeated indices are summed). The matrix g_ij is symmetric because an inner product is, and positive-definite because g_p(v, v) > 0 for every nonzero v — that positivity is what makes lengths real and nonnegative.
Two concrete cases anchor the formalism. On flat R^2 the metric is the constant identity, g_ij = [1, 0; 0, 1] in MATLAB-style notation, so ds^2 = dx^2 + dy^2 — Pythagoras, recovered. On the unit round sphere S^2 in spherical coordinates the metric is ds^2 = d(theta)^2 + sin^2(theta) d(phi)^2: notice the g_ij now genuinely depend on the point (the sin^2(theta) factor), which is exactly the curving you cannot see on flat R^2. That position-dependence of g_ij is where all the interesting geometry will eventually come from.
From metric to geometry: lengths, angles, and volume
Now watch the inner product do real work. The length of a tangent vector v is |v| = sqrt(g_p(v, v)), and the angle between v and w is read off from g_p(v, w) = |v| |w| cos(theta), exactly as in Euclidean space — because pointwise that is precisely what a positive-definite inner product gives you. To measure a whole curve c(t) from a to b, integrate the speed of its velocity: length = integral from a to b of sqrt(g(c'(t), c'(t))) dt. That single integral is the bridge from infinitesimal data (a metric) to a global number (a length), and it is the seed of geodesics in Guide 3 — the curves that locally minimize it.
The metric also fixes how much space occupies a region. In coordinates the Riemannian volume form is dV = sqrt(det(g_ij)) dx^1 ^ ... ^ dx^n: the naive coordinate volume corrected by the factor sqrt(det g), which records how the metric stretches or squeezes the coordinate grid. On flat R^n with g_ij = identity this factor is 1 and you recover ordinary volume; on the round sphere the sin(theta) in the metric becomes the familiar sin(theta) area weight. If you have met the first fundamental form of a surface in Volume I, a Riemannian metric is precisely the intrinsic, all-dimensions generalization of that idea — the first fundamental form WAS a metric, you simply did not call it one.
Every manifold admits a metric — and why that is non-obvious
Does such a g even exist on a given manifold? Locally yes, trivially: in any single chart just copy the Euclidean inner product, declaring g_ij = the identity matrix. The genuine question is global — can the local pieces be glued into one smooth metric defined everywhere at once? The answer is yes for every smooth manifold (assuming the mild, standard hypothesis that M is second-countable / paracompact), and the tool that performs the gluing is the partition of unity: a family of bump functions subordinate to a chart cover that sum to 1, letting you average the local Euclidean metrics into a single global one.
The averaging trick works precisely because of positive-definiteness: a sum of positive-definite matrices weighted by nonnegative bump functions (that do not all vanish at any point) is again positive-definite, so the patched-together g stays a legitimate metric. This is worth savoring — it would FAIL for Lorentzian metrics, which can obstruct globally and need topological conditions to exist. So 'a manifold has a Riemannian metric' is automatic, but it is automatic for a real reason, not by decree. Henceforth we may freely assume M comes equipped with a g and call the pair (M, g) a Riemannian manifold.
The fundamental theorem: a metric picks one connection
Here is the problem the whole subject pivots on. A metric lets you compare vectors at a single point, but to do calculus — to differentiate a vector field, to ask whether a curve bends — you must compare vectors at NEARBY points, and a bare manifold gives no canonical way to do that. The structure that supplies it is a connection nabla, a rule taking a vector field X and a vector field Y to a new field nabla_X Y, the covariant derivative of Y in the direction X. Guide 2 develops connections in full; the miracle, stated now, is that a metric selects one. There are infinitely many connections on a manifold, but the metric's geometry forces a unique best choice.
The selection is made by two conditions. First, metric compatibility: nabla should respect g, meaning the product rule X(g(Y, Z)) = g(nabla_X Y, Z) + g(Y, nabla_X Z) holds — differentiating lengths and angles is consistent with parallel transport, so vectors keep their length as you carry them along. Second, torsion-freeness (symmetry): nabla_X Y - nabla_Y X = [X, Y], where [X, Y] is the Lie bracket you met in differential topology — the connection adds no spurious twist beyond the manifold's own bracket. The Fundamental Theorem of Riemannian Geometry states that these two reasonable demands are satisfied by exactly one connection.
Why it is forced: the Koszul formula and Christoffel symbols
Uniqueness is not a mystery — you can corner the connection algebraically. Write the compatibility identity three times, cyclically permuting the roles of X, Y, Z, then add two of the equations and subtract the third. Every term involving nabla cancels except one, leaving the Koszul formula, which expresses 2 g(nabla_X Y, Z) entirely in terms of the metric g, directional derivatives of g, and Lie brackets — no nabla on the right at all. Since g is nondegenerate, knowing g(nabla_X Y, Z) for all Z determines nabla_X Y completely. That single algebraic squeeze proves both halves at once: existence (the formula defines a connection) and uniqueness (any compatible torsion-free nabla must obey it).
In coordinates the unique connection becomes the Levi-Civita connection, and its components are the Christoffel symbols Gamma^k_ij, defined by nabla_{d/dx^i}(d/dx^j) = Gamma^k_ij d/dx^k. The Koszul formula, written in a coordinate frame where all Lie brackets vanish, collapses to an explicit recipe: Gamma^k_ij is built from first derivatives of the metric entries and the inverse matrix g^{kl}. You do not have to take this on faith — it is a finite computation you will run by hand for the sphere and the hyperbolic plane in Guide 2. Two honest warnings: the Gamma^k_ij are NOT the components of a tensor (they transform with an inhomogeneous correction term under change of coordinates), and the sign and index conventions differ across textbooks, so always fix yours before quoting a formula.
Koszul (defines the unique nabla):
2 g( nabla_X Y , Z ) = X g(Y,Z) + Y g(Z,X) - Z g(X,Y)
+ g([X,Y],Z) - g([Y,Z],X) + g([Z,X],Y)
Christoffel symbols (coordinate version):
Gamma^k_ij = (1/2) g^{kl} ( d_i g_{jl} + d_j g_{il} - d_l g_{ij} )
where d_i = d/dx^i and g^{kl} = inverse of g_{ij}Where this leaves you, and where the rung goes next
Step back and see what you have bought with one inner product per point. The metric g gave you lengths, angles, and volume; the Fundamental Theorem then handed you, for free and uniquely, a way to differentiate vector fields — the Levi-Civita connection nabla. Everything in the rest of Volume II's geometry rests on this pairing. Geodesics are curves with nabla_{c'} c' = 0 (zero acceleration); parallel transport is solving nabla = 0 along a curve; and the Riemann curvature tensor R(X, Y)Z = nabla_X nabla_Y Z - nabla_Y nabla_X Z - nabla_{[X,Y]} Z measures the failure of second covariant derivatives to commute. The connection you just learned exists is the single object all of those are built from.
- Guide 2 unpacks the connection nabla and computes the Christoffel symbols Gamma^k_ij explicitly for the sphere and the hyperbolic plane, then defines covariant differentiation of any tensor.
- Guide 3 turns nabla into geodesics, the exponential map, and normal coordinates — the curved analogue of straight lines and the cleanest coordinates near a point.
- Guide 4 builds the Riemann curvature tensor from nabla and distills it into sectional curvature K, Ricci curvature Ric, and scalar curvature S.
- Guide 5 returns to submanifolds: the second fundamental form, the Gauss-Codazzi equations, and the constant-curvature space forms (sphere, Euclidean space, hyperbolic space).