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For a smooth map $f:X\to Y$ between compact connected $n$-manifolds and a regular value $y$, defineThe regular-value theorem makes the preimage discrete and compact, hence finite. A transverse path between regular values produces a compact one-manifold whose boundary is the two fibres, proving independence of $y$. The degree is invariant under smooth homotopy.

A value is regular when the derivative is surjective at every point of its preimage.

Every value of a submersion is regular, but a map can have some regular values without being a submersion everywhere.

A chart is a homeomorphism from an open subset of a manifold to an open subset of Euclidean space.

A cone is intrinsically flat away from its vertex but has nontrivial angular holonomy.

A ruled surface is swept out by a smoothly varying one-parameter family of affine lines.

A helicoid is a ruled minimal surface swept out by a line that rotates while translating along its perpendicular axis.

A surface of revolution is invariant under rotations around a fixed axis and is locally generated by rotating a profile curve.

An embedded-surface chart is a homeomorphic smooth parametrisation with derivative of rank two.

Riemannian geometry studies smooth manifolds equipped with a Riemannian metric: an inner product at each tangent space, varying smoothly with position. This local measuring rule determines angles, arc length, Riemannian distance and geodesics. An inhabitant of a surface can discover its geometry by measuring within it, without knowing how it sits in a larger Euclidean space.

A circular cylinder is a particularly useful example. Its surface bends in space, yet unrolling it preserves the induced metric: a small patch has the geometry of a flat sheet. By contrast, a round sphere has positive Gaussian curvature. The distinction between intrinsic flatness of a circular cylinder and extrinsic curvature separates the geometry an inhabitant measures from the way an outside observer sees the surface bend.

Geodesics supply the analogue of straight lines. On a round sphere they follow great circles; on a circular cylinder they become straight lines when lifted to an unrolled sheet. Their arc length lets us compare candidate routes, as in great-circle distance and Riemannian distance on a circular cylinder. The same measuring rule also defines differential operators such as the Laplace-Beltrami operator, connecting geometry with diffusion and spectral geometry.

Bernhard Riemann's 1854 inaugural lecture proposed studying spaces of arbitrary dimension through their infinitesimal measuring rules, including a positive quadratic expression for the squared line element. This moved the subject beyond a particular surrounding space: the measuring rule itself became part of the geometry. ^[ref].

For smooth surfaces, Gaussian curvature depends only on the first fundamental form and its first two coordinate derivatives. Consequently any smooth local isometry $f:M\to N$ satisfies $K_N(f(p))=K_M(p)$. The surprise is that $K$ is defined using the shape operator, which describes bending in space, yet its value can be recovered from measurements made entirely within the surface.

In his 1827 paper, Carl Friedrich Gauss expressed Gaussian curvature through the metric coefficients and their partial derivatives in Article 11, then stated its invariance under development in Article 12. Article 13 explicitly compares a plane with surfaces developable onto it, including cylindrical surfaces. The paper was presented on 8 October 1827 and printed in 1828; the Morehead–Hiltebeitel English translation appeared in 1902. ^[ref]

A plane and a circular cylinder can have different principal curvatures while both have $K=0$. Their matching first fundamental forms exhibit the actual local isometry. In contrast, no nonempty open patch of a sphere of radius $R$ can be flattened smoothly into a plane while preserving its first fundamental form: $1/R^2$ cannot become zero. Making the patch small reduces practical distortion, but never produces an exact local isometry. This is a geometric obstruction to a perfectly distance-preserving flat map of a spherical world, independent of where one places a seam.

The following proof outline connects the two descriptions of Gaussian curvature. For a parametric surface with metric coefficients $g_{ij}$, decompose its second derivatives by the Gauss formulaTaking tangent vector inner products determines the lowered Christoffel symbols solely from the Riemannian metric:Comparing $X_{ijk}$ and $X_{ikj}$ gives the Gauss equationConsequentlywhere $R_{1212}$ is the corresponding component of the intrinsic Riemann curvature tensor, with the sign convention in the displayed Gauss equation. The final expression uses only $g_{ij}$ and its first two derivatives.

The inverse image of a regular value of a smooth map is a submanifold of codimension equal to the target dimension.

For a unit-speed curve $\alpha$ on an oriented surface with unit normal $N$, let $T=\dot{\alpha }$ and let $D_s$ denote tangential covariant differentiation. Its signed geodesic curvature isIt vanishes exactly when the curve is a geodesic.

For a unit-speed space curve with nonzero curvature, $t=\dot{\alpha }$, $\kappa =|\dot{t}|$, $n=\dot{t}/\kappa$, and $b=t\times n$. With the torsion convention used here,

Steiner symmetrization replaces every chord perpendicular to a chosen axis by a centered chord of the same length.

For unit speed, $\kappa =|T^{\prime }|$ and $\tau =-B^{\prime }\cdot N$. Generally $\kappa =|r^{\prime }\times r^{\prime \prime }|/|r^{\prime }{|}^3$ and $\tau =\det (r^{\prime },r^{\prime \prime },r^{\prime \prime \prime })/|r^{\prime }\times r^{\prime \prime }{|}^2$.

A smooth curve is a smooth map from an interval into a manifold or Euclidean space.

A curve may be described by a parametric equation; smoothness requires its coordinate functions to be differentiable to all orders.

For an oriented regular surface in ${\mathbb{R}}^3$, the Gauss map assigns its chosen unit normal

The first fundamental form of a smooth surface $S\subseteq {\mathbb{R}}^3$ is the restriction of the ambient inner product to each tangent space $T_{p}S$. For a regular parametric surface $X(u,v)$, it is

Here $X_u^2$ means $X_u\cdot X_u$, not a componentwise square. The coefficients form the Gram matrix of the two coordinate tangent vectors. Regularity makes this a positive-definite matrix, so $E>0$ and $EG-F^2>0$.

To calculate the length of a curve $\gamma (t)=X(u(t),v(t))$, apply the chain rule and integrate its speed:The same inner product gives the angle $\alpha$ between nonzero tangent vectors $V,W$ through $\cos \alpha =I(V,W)/\sqrt{I(V,V)I(W,W)}$, and gives the area element $dA=\sqrt{EG-F^2}dudv$ as a positive density. One compact expression therefore supplies rulers, protractors, and surface area measurements.

For the lateral surface of a circular cylinder, take the circumferential arc length $u$ rather than the angle:Direct inner products give $E=G=1,F=0$, hence $I=d{u}^2+d{v}^2$. This is exactly the Euclidean metric on the unrolled plane. For instance, a path with $u=3t,v=4t$, $0\le t\le 1$, has length $5$ on either surface. Every corresponding path retains its length, not merely this example, so sufficiently small neighborhoods retain their intrinsic distances. The local isometry preserves angles and surface areas too. Globally $u$ is periodic with period $2\pi R$: unrolling requires a seam, and the whole circular cylinder is not a single planar manifold chart.

An orientation of a regular surface is a continuous choice of unit normal. Reversing orientation replaces $N$ by $-N$.