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For a smooth function $f(x,y)$,where the smooth remainder is

The Grassmannian $\mathrm{Gr}(k,V)$ parametrizes the $k$-dimensional linear subspaces of a finite-dimensional vector space $V$. Its Plücker embedding makes it a projective algebraic variety.

The real symplectic group consists of matrices $A$ satisfying $A^{T}JA=J$. It is an embedded matrix Lie group of dimension $2n^2+n$ in size $2n$.

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 flat cone of total cone angle $\theta >0$ has metricwith every point at $r=0$ identified to its vertex. Away from the vertex, $t=\theta \phi /(2\pi )$ unfolds the metric locally into the Euclidean plane, so Gaussian curvature is zero there.

The angular defect at the vertex is $2\pi -\theta$. Parallel transport once around it rotates a tangent vector by this defect modulo $2\pi$. This holonomy can be trivial: angle $4\pi$ still gives a singular vertex but zero rotational holonomy modulo $2\pi$. The angle $2\pi$ gives the smooth Euclidean plane. A punctured circular cone is an embedded example with its vertex removed.

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 compact oriented Riemannian surface $D$ with piecewise smooth boundary,Here $K$ is Gaussian curvature, $dA$ the area element of a surface, $k_g$ signed geodesic curvature, and $\chi (D)$ the Euler characteristic. Traverse each boundary component with the surface on the left. The signed exterior angle at a convex corner of interior angle ${\alpha }_j$ is ${\epsilon }_j=\pi -{\alpha }_j$. Empty boundaries contribute no boundary or corner terms.^[ref]

On a disk in the Euclidean plane, zero interior Gaussian curvature leaves total boundary turning $2\pi$. On the unit sphere, $K=1$ and $\chi =2$, giving area $4\pi$. A geodesic triangle has no smooth boundary term, giving the spherical excess formula or hyperbolic angle defect.

For a closed surface, a local metric quantity integrates to a topological invariant. A proof triangulates the surface and applies the local turning formula to each triangle. Shared-edge contributions cancel and the angle sums leave $2\pi (V-E+F)$. Thus changing the Riemannian metric cannot change total Gaussian curvature.^[ref]

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 about a line replaces every perpendicular slice of a measurable set by a centered interval of the same one-dimensional Lebesgue measure. For a bounded Borel set $E\subset {\mathbb{R}}^2$, take the horizontal axis as the symmetry line and putSlice endpoints do not affect area. A slice can consist of several disjoint intervals: its total length determines the replacement.^[ref]

For a convex set bounded by graphs $u_{-}(x)$ and $u_{+}(x)$, the new bounds are $\pm (u_{+}-u_{-})/2$. A disk centered on the symmetry axis stays unchanged. Fubini's theorem proves $\mathrm{area}(E^{\ast })=\int {\ell }_E(x)dx=\mathrm{area}(E)$. Preserved area together with perimeter decrease under Steiner symmetrization makes the operation useful for the isoperimetric inequality.

The curvature of a space curve measures turning of its unit tangent vector; the torsion of a curve measures twisting of its osculating plane. For a regular curve $r(t)$ in Euclidean space ${\mathbb{R}}^3$,The first formula requires two derivatives and $r^{\prime }\ne 0$. The second requires three derivatives and $r^{\prime }\times r^{\prime \prime }\ne 0$: torsion of a curve is undefined by this formula at zero curvature of a space curve.

With arc length $s$ and Frenet frame $(T,N,B)$, the formulas become $\kappa =\Vert dT/ds\Vert$ and $\tau =-⟨dB/ds,N⟩$. A circle of radius $a>0$ has $\kappa =1/a$ and $\tau =0$. The circular helix $r(t)=(a\cos t,a\sin t,bt)$ has $\kappa =a/(a^2+b^2)$ and $\tau =b/(a^2+b^2)$. These data, expressed in arc length, determine the curve up to a proper Euclidean motion of Euclidean three-space by the fundamental theorem of regular space curves.^[ref]

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.