Intrinsic metrics and local isometries

Try

Cut a cylinder along one vertical generator and unroll it. In coordinates u measuring circumference and v measuring height, its first form is du²+dv². Every curve contained in a suitable patch has the same length before and after this local unrolling.

The full wrapping map from the entire plane is different. Points separated by 2πR in the u direction have the same image. A map can preserve tangent lengths locally while failing to preserve global point-to-point distances or injectivity.

Predict

Predict whether a local isometry must preserve the shape operator. The plane and cylinder have different principal curvatures, so it cannot. Gaussian curvature is preserved by the intrinsic Gauss theorem developed later.

Predict whether an angle-preserving map is necessarily an isometry. Uniform scaling preserves angles but multiplies lengths. A variable positive conformal scale preserves pointwise angles while changing lengths along paths.

Experiment

Separate three preservation claims.

Map or metric change	Angles	Local lengths	Global injectivity
Plane to a short cylinder patch	Preserved	Preserved	Yes on that patch
Full plane wrapping onto cylinder	Preserved locally	Preserved locally	No
Metric multiplied by e^(2φ)	Preserved at each point	Generally changed	Not the relevant question

Choose two plane points a full circumference apart and verify their images coincide. This is an exact obstruction to global isometry even though the pullback metric calculation is correct everywhere.

Understand

A local diffeomorphism F between Riemannian surfaces is a local isometry when ⟨dF(v),dF(w)⟩ at the image equals ⟨v,w⟩ at the source for every tangent pair. By the chain rule it preserves curve speeds and hence lengths for curves in its domain. The cylinder wrapping map has this property because its coordinate derivatives are orthonormal.

For a conformal metric g=e^(2φ)(dx²+dy²), the angle numerator and the product of vector norms both acquire e^(2φ). Their quotient is unchanged, proving angle preservation. A curve’s speed instead acquires e^φ, whose integral generally changes. Conformality is therefore weaker than isometry.

Intrinsic distance is the infimum of lengths over all connecting surface paths. A local isometry preserves sufficiently local distance behavior, but a global shortcut or identification can change the infimum over all paths. This explains why a pullback identity alone does not prove a global metric-space isometry.

Apply

Repair “Equal Gaussian curvature proves global isometry.” Even once K is known intrinsic, matching one scalar function does not generally determine a global metric. A plane and a cylinder both have K=0 but differ globally. Local constant-curvature classification requires a separate theorem, and variable curvature data can be subtler still.

For optional depth, a Riemannian metric can be defined abstractly by smooth positive-definite bilinear forms without a preferred embedding. Coordinate transformation rules keep the definition consistent across charts. Connections, geodesics and intrinsic curvature can then be constructed from the metric alone; principal curvatures require an embedding and normal.

The immediate goal is to derive that metric connection rather than rely on the visual intuition of unrolling.

Continue

When claiming two surfaces are equivalent, specify local or global, angle or length, and whether an embedding must be preserved. Continue to Christoffel symbols and the connection to differentiate tangent fields intrinsically.