the fundamental theorem of Riemannian geometry
A metric tells you lengths and angles, but to do calculus on a curved space you also need to differentiate vector fields — to compare a vector at one point with a vector at a nearby point. There are many ways to set up such a derivative (a connection), and a priori the metric does not single one out. The fundamental theorem says it does, uniquely, once you ask for two natural compatibility conditions: out of all connections, exactly one respects the metric and has no twisting.
Precisely: on any Riemannian manifold (M, g) there exists a unique affine connection nabla, called the Levi-Civita connection, that is (1) metric-compatible, meaning nabla g = 0, equivalently X(g(Y,Z)) = g(nabla_X Y, Z) + g(Y, nabla_X Z) so parallel transport preserves inner products; and (2) torsion-free (symmetric), meaning nabla_X Y - nabla_Y X = [X, Y]. Existence and uniqueness follow from the Koszul formula, which solves for nabla explicitly: 2 g(nabla_X Y, Z) = X g(Y,Z) + Y g(X,Z) - Z g(X,Y) + g([X,Y],Z) - g([X,Z],Y) - g([Y,Z],X). The right side uses only g and Lie brackets, so nabla is forced.
Why this is foundational: it means the metric ALONE determines how to differentiate, hence geodesics, curvature, and all intrinsic geometry follow from g with no further choices. It is the rigorous engine behind Gauss's theorema egregium generalized to all dimensions. Be honest about the hypotheses though — drop torsion-freeness and you get a whole world of metric connections with torsion (used in some physics); drop metric-compatibility and you lose the link between the connection and lengths. The theorem's force is precisely that BOTH conditions together pin down exactly one connection.
Take ordinary R^n with the flat metric. The Koszul formula gives nabla_X Y = the ordinary directional derivative (all Christoffel symbols vanish in standard coordinates), so the unique Levi-Civita connection on flat space is just the usual calculus derivative — exactly what you would hope.
On flat space the unique metric-compatible torsion-free connection reduces to ordinary differentiation.
Uniqueness needs BOTH metric-compatibility and torsion-freeness; either condition alone leaves infinitely many connections. The theorem is not that 'a metric gives a connection' loosely, but that these two specific axioms determine exactly one.