metric tensor
Geometry begins when you can measure: how long is this arrow, what angle do these two directions make, how far apart are these two nearby points. On a flat sheet with Cartesian axes those questions have the familiar Pythagorean answers. But on a sphere, in polar coordinates, or in curved spacetime, the relationship between coordinate differences and actual distances is no longer simple — and the metric tensor is the object that stores that relationship at every point.
Concretely the metric is a symmetric rank-2 covariant tensor g_{ij}. Its central job is to turn coordinate displacements into real squared lengths through the line element ds^2 = g_{ij} dx^i dx^j. The diagonal entries g_{ii} stretch each coordinate direction, the off-diagonal entries g_{ij} record how non-perpendicular the coordinate axes are. More generally g defines the inner product of any two vectors, u dot v = g_{ij} u^i v^j, so it delivers both lengths and angles. In flat Cartesian space g_{ij} is just the identity (so ds^2 = dx^2 + dy^2 + dz^2); in 2-D polar coordinates it is the diagonal [1, 0; 0, r^2], which is why an angular step d(theta) contributes a true arc length of r d(theta), not just d(theta). The inverse matrix g^{ij} exists wherever g is non-degenerate, and the pair g_{ij}, g^{ij} is the machinery that lowers and raises indices.
The metric is the single most important object in differential geometry and general relativity. Once you have g you can compute lengths of curves, areas, angles, the Christoffel symbols, geodesics, and ultimately the curvature — everything geometric is downstream of the metric. In Einstein's theory the metric of spacetime IS the gravitational field: matter and energy tell the metric how to curve (the Einstein field equations), and the curved metric tells matter how to move along geodesics. A subtlety worth stating honestly: in relativity the metric is not positive-definite but has mixed signs (one time, three space), so ds^2 can be positive, negative, or zero — distinguishing time-like, space-like, and light-like separations.
On the surface of a sphere of radius a, using latitude-like angle theta and longitude phi, the metric is g = [a^2, 0; 0, a^2 sin^2(theta)], so ds^2 = a^2 d(theta)^2 + a^2 sin^2(theta) d(phi)^2. The sin^2(theta) factor is geometry talking: a step in longitude near the equator (theta = 90 degrees) covers real distance, but the same step near the pole (theta near 0) covers almost none.
The sphere's metric encodes that meridians crowd together near the poles — pure geometry, no embedding needed.
A non-identity metric does not by itself mean the space is curved: flat space in polar or spherical coordinates has a non-trivial g but zero curvature. Curvature is detected by second derivatives of g (the Riemann tensor), not by g looking complicated.