First fundamental form ($I$)

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.

A local isometry between Riemannian manifolds is a local diffeomorphism $f$ with $f^{\ast }{g}_N=g_M$. Equivalently, its differentials are isometric linear isomorphisms between tangent spaces. It preserves the Levi-Civita connection, intrinsic sectional curvature, and affinely parametrized geodesics. A map whose differentials are merely isometric injections is an isometric immersion; the local diffeomorphism requirement distinguishes these notions.

The helicoid $H(s,v)=(\sinh s\cos v,\sinh s\sin v,v)$ and catenoid $C(s,v)=(\cosh s\cos v,\cosh s\sin v,s)$ have the same first fundamental form ${\cosh }^{2}s(d{s}^2+d{v}^2)$. Therefore matching equal parameter pairs gives a local isometry. The result is local: a full turn changes the height on the helicoid while the angular coordinate on the catenoid repeats. Equal intrinsic lengths do not require the same embedding into Euclidean space.

Two local isometries from a connected Riemannian manifold that have the same value and differential at one point agree everywhere. They preserve geodesics, so their equality follows on a normal neighborhood from $f({\exp }_{p}v)={\exp }_{f(p)}(d{f}_{p}v)$. The set of points where both value and differential agree is consequently open, and continuity makes it closed. Connectedness completes the argument.

A local isometry from a nonempty complete Riemannian manifold to a connected Riemannian manifold is a surjective covering map. Lift radial geodesics in a normal ball using completeness; the exponential map at each inverse-image centre maps a tangent ball diffeomorphically onto a sheet. Uniqueness of the geodesic initial-value problem makes these sheets disjoint and exhaustive. Surjectivity follows by lifting piecewise geodesic paths. Conversely a Riemannian covering of a complete target is complete, by lifting complete target geodesics and using the Hopf-Rinow theorem. If the target is disconnected, the first assertion gives a covering only of the union of components met by the map.

A local isometry intertwines the Levi--Civita covariant derivatives. It therefore sends every affinely parametrized geodesic to an affinely parametrized geodesic.

A nonunit Euclidean dilation sends straight-line geodesics to straight-line geodesics but scales their tangent inner products. Preservation of geodesics alone therefore does not imply local isometry.

A Riemannian isometry is a diffeomorphism that preserves the Riemannian metric, and hence lengths, areas, and intrinsic curvature.

An isometry between oriented Riemannian manifolds that reverses orientation. Its differential is orthogonal with determinant minus one in oriented orthonormal frames, so it reverses the oriented volume form while preserving the associated positive volume measure. Reflection in a geodesic of the hyperbolic plane is an example. An orientation-reversing isometry still identifies the scalar Laplace spectra, even though it does not preserve the chosen complex orientation of a Riemann surface.

For the conethe first fundamental form isThe local change of polar coordinatesturns this into the Euclidean metric $d{R}^2+R^{2}d{\Theta }^2$.

For a circular cone with intrinsic metric $d{\rho }^2+{\lambda }^2{\rho }^{2}d{\theta }^2$, $0<\lambda \le 1$, local development uses plane angle $\varphi =\lambda \theta$. A plane straight segment that avoids the origin lifts to a geodesic. Choosing an angular difference at most $\pi$ gives plane difference at most $\pi \lambda$, so for $\lambda <1$ this constructs a joining geodesic between any two distinct points. Different permitted angle lifts can yield distinct joining geodesics; the missing apex cannot be used to join straight segments through the plane origin.

In a positively oriented surface chart,The coordinate-change Jacobian and the metric determinant transform inversely, so this defines a global area form.

The surface area of a measurable region $U\subseteq S$ is ${\int }_{U}dA$. For a parametrization $X(u,v)$ it is $\int |X_u\times X_v|dudv$.

For projection of a planar region with unit normal $\mathbf{n}$ onto a plane with unit normal $\mathbf{k}$, the projected area is $|\mathbf{n}\cdot \mathbf{k}|$ times the original area. Indeed, the signed projected area of the parallelogram spanned by tangent vectors $\mathbf{u},\mathbf{v}$ is $(\mathbf{u}\times \mathbf{v})\cdot \mathbf{k}$, while $\mathbf{u}\times \mathbf{v}$ is parallel to $\mathbf{n}$. Taking magnitudes proves the factor. The same pointwise formula applies to a parametrized surface; adding projected patches counts multiplicity unless their images overlap only in sets of zero area.

The area of a planar oriented region with a sign determined by orientation. For a triangle with cyclically ordered points $A,B,C$, it is $\frac{1}{2}((B-A)\times (C-A))$, using the two-dimensional cross product; reversing the order reverses the sign.

For an oriented parametrized surface $\mathbf{r}(u,v)$,

For a regular implicit surface $f(x,y,z)=0$ with $f_z\ne 0$, the implicit function theorem gives a local graph $z=h(x,y)$. Its upward vector area element is $(-h_x,-h_y,1)dxdy$. Differentiating $f(x,y,h(x,y))=0$ yields $h_x=-f_x/f_z$, $h_y=-f_y/f_z$, proving the formula. Division by $f_z$ makes the final component positive independently of the sign of the original gradient. At a point where $f_z=0$, another projection or parametrization is needed.

The vector area of an oriented surface is the integral of its unit normal vector times its area element. If its oriented boundary is a closed curve $C$, Stokes theorem givesConsequently it depends only on the oriented boundary, not on the spanning surface. For a planar surface it is signed area times a unit normal vector. A current loop has magnetic dipole moment $\mathbf{m}=I\mathbf{S}$; the scalar area of a nonplanar spanning surface is not interchangeable with this vector.

For the faces of a closed polyhedron, take outward unit normal vectors. Applying the divergence theorem to an arbitrary constant vector field $c$ givesSince $c$ is arbitrary, the vector areas sum to zero. This is an oriented statement; the positive scalar surface areas do not cancel. The same proof works for every piecewise smooth closed boundary to which the divergence theorem applies.

For an oriented closed polygon, triangulate with a common vertex $P$. Its total vector area is $\frac{1}{2}{\sum }_i(B_i-P)\times (B_{i+1}-P)$; the terms involving $P$ telescope, giving the displayed boundary-only formula. Its signed projected area along a unit vector $V$ is $V\cdot \mathbf{A}$. The maximum is $|\mathbf{A}|$, in direction $\mathbf{A}/|\mathbf{A}|$ if nonzero, and zero projections have directions perpendicular to $\mathbf{A}$. If $\mathbf{A}=0$, every direction has zero signed projection despite possible nonzero unsigned surface area.