Definition

It is intuitively clear that a sphere is smooth, while a cone or a pyramid, due to their vertex or edges, are not. The notion of a "regular surface" is a formalization of the notion of a smooth surface. The definition utilizes the local representation of a surface via maps between Euclidean spaces. There is a standard notion of smoothness for such maps; a map between two open subsets of Euclidean space is smooth if its partial derivatives of every order exist at every point of the domain.^[6][7][8]

The following gives three equivalent ways to present the definition; the middle definition is perhaps the most visually intuitive, as it essentially says that a regular surface is a subset of ℝ³ which is locally the graph of a smooth function (whether over a region in the yz plane, the xz plane, or the xy plane).

A local parametrization of Monge patch form for the upper hemisphere of the 2-sphere, obtained by projecting onto the xy-plane

The homeomorphisms appearing in the first definition are known as local parametrizations or local coordinate systems or local charts on S.^[13] The equivalence of the first two definitions asserts that, around any point on a regular surface, there always exist local parametrizations of the form (u, v) ↦ (h(u, v), u, v), (u, v) ↦ (u, h(u, v), v), or (u, v) ↦ (u, v, h(u, v)), known as Monge patches. Functions F as in the third definition are called local defining functions. The equivalence of all three definitions follows from the implicit function theorem.^[14][15][16]

Coordinate changes between different local charts must be smooth

Given any two local parametrizations f : V → U and f ′ : V ′→ U ′ of a regular surface, the composition f ⁻¹ ∘ f ′ is necessarily smooth as a map between open subsets of ℝ².^[17] This shows that any regular surface naturally has the structure of a smooth manifold, with a smooth atlas being given by the inverses of local parametrizations.

In the classical theory of differential geometry, surfaces are usually studied only in the regular case.^[7][18] It is, however, also common to study non-regular surfaces, in which the two partial derivatives ∂_u f and ∂_v f of a local parametrization may fail to be linearly independent. In this case, S may have singularities such as cuspidal edges. Such surfaces are typically studied in singularity theory. Other weakened forms of regular surfaces occur in computer-aided design, where a surface is broken apart into disjoint pieces, with the derivatives of local parametrizations failing to even be continuous along the boundaries.

A hyperboloid of two sheets

A torus

A helicoid

Simple examples. A simple example of a regular surface is given by the 2-sphere {(x, y, z) | x² + y² + z² = 1}; this surface can be covered by six Monge patches (two of each of the three types given above), taking h(u, v) = ± (1 − u² − v²)^1/2. It can also be covered by two local parametrizations, using stereographic projection. The set {(x, y, z) : ((x² + y²)^1/2 − r)² + z² = R²} is a torus of revolution with radii r and R. It is a regular surface; local parametrizations can be given of the form

${\displaystyle f(s,t)={\big (}(R\cos s+r)\cos t,(R\cos s+r)\sin t,R\sin s{\big )}.}$

The hyperboloid on two sheets {(x, y, z) : z² = 1 + x² + y²} is a regular surface; it can be covered by two Monge patches, with h(u, v) = ±(1 + u² + v²)^1/2. The helicoid appears in the theory of minimal surfaces. It is covered by a single local parametrization, f(u, v) = (u sin v, u cos v, v).

Tangent vectors and normal vectors

Let S be a regular surface in ℝ³, and let p be an element of S. Using any of the above definitions, one can single out certain vectors in ℝ³ as being tangent to S at p, and certain vectors in ℝ³ as being orthogonal to S at p.

Objects used in definition	A vector X in ℝ3 is tangent to S at p if...	A vector n in ℝ3 is normal to S at p if...
Local parametrizations	... given any local parametrization f : V → S with p ∈ f(V), X is a linear combination of ${\displaystyle \textstyle {\frac {\partial f}{\partial u}}{\big |}_{f^{-1}(p)}}$ and ${\displaystyle \textstyle {\frac {\partial f}{\partial v}}{\big |}_{f^{-1}(p)}}$	... it is orthogonal to every tangent vector to S at p
Monge patches	... for any Monge patch (u, v) ↦ (u, v, h(u, v)) whose range includes p, one has ${\displaystyle X^{1}{\frac {\partial h}{\partial u}}+X^{2}{\frac {\partial h}{\partial v}}-X^{3}=0,}$ with the partial derivatives evaluated at the point (p1, p2). The analogous definition applies in the case of the Monge patches of the other two forms.	... for any Monge patch (u, v) ↦ (u, v, h(u, v)) whose range includes p, n is a multiple of (∂h/∂u, ∂h/∂v, −1) as evaluated at the point (p1, p2). The analogous definition applies in the case of the Monge patches of the other two forms.
Implicit functions	... for any local defining function F whose domain contains p, X is orthogonal to ∇F(p)	... for any local defining function F whose domain contains p, n is a multiple of ∇F(p)

One sees that the tangent space or tangent plane to S at p, which is defined to consist of all tangent vectors to S at p, is a two-dimensional linear subspace of ℝ³; it is often denoted by T_pS. The normal space to S at p, which is defined to consist of all normal vectors to S at p, is a one-dimensional linear subspace of ℝ³ which is orthogonal to the tangent space T_pS. As such, at each point p of S, there are two normal vectors of unit length (unit normal vectors). The unit normal vectors at p can be given in terms of local parametrizations, Monge patches, or local defining functions, via the formulas

${\displaystyle \pm \left.{\frac {{\frac {\partial f}{\partial u}}\times {\frac {\partial f}{\partial v}}}{{\big \|}{\frac {\partial f}{\partial u}}\times {\frac {\partial f}{\partial v}}{\big \|}}}\right|_{f^{-1}(p)},\qquad \pm \left.{\frac {{\big (}{\frac {\partial h}{\partial u}},{\frac {\partial h}{\partial v}},-1{\big )}}{\sqrt {1+{\big (}{\frac {\partial h}{\partial u}}{\big )}^{2}+{\big (}{\frac {\partial h}{\partial v}}{\big )}^{2}}}}\right|_{(p_{1},p_{2})},\qquad {\text{or}}\qquad \pm {\frac {\nabla F(p)}{{\big \|}\nabla F(p){\big \|}}},}$

following the same notations as in the previous definitions.

It is also useful to note an "intrinsic" definition of tangent vectors, which is typical of the generalization of regular surface theory to the setting of smooth manifolds. It defines the tangent space as an abstract two-dimensional real vector space, rather than as a linear subspace of ℝ³. In this definition, one says that a tangent vector to S at p is an assignment, to each local parametrization f : V → S with p ∈ f(V), of two numbers X¹ and X², such that for any other local parametrization f ′ : V → S with p ∈ f(V) (and with corresponding numbers (X ′)¹ and (X ′)²), one has

${\displaystyle {\begin{pmatrix}X^{1}\\X^{2}\end{pmatrix}}=A_{f'(p)}{\begin{pmatrix}(X')^{1}\\(X')^{2}\end{pmatrix}},}$

where A_{f ′(p)} is the Jacobian matrix of the mapping f ⁻¹ ∘ f ′, evaluated at the point f ′(p). The collection of tangent vectors to S at p naturally has the structure of a two-dimensional vector space. A tangent vector in this sense corresponds to a tangent vector in the previous sense by considering the vector

${\displaystyle X^{1}{\frac {\partial f}{\partial u}}+X^{2}{\frac {\partial f}{\partial v}}.}$

in ℝ³. The Jacobian condition on X¹ and X² ensures, by the chain rule, that this vector does not depend on f.

For smooth functions on a surface, vector fields (i.e. tangent vector fields) have an important interpretation as first order operators or derivations. Let ${\displaystyle S}$ be a regular surface, ${\displaystyle U}$ an open subset of the plane and ${\displaystyle f:U\rightarrow S}$ a coordinate chart. If ${\displaystyle V=f(U)}$, the space ${\displaystyle C^{\infty }(U)}$ can be identified with ${\displaystyle C^{\infty }(V)}$. Similarly ${\displaystyle f}$ identifies vector fields on ${\displaystyle U}$ with vector fields on ${\displaystyle V}$. Taking standard variables u and v, a vector field has the form ${\displaystyle X=a\partial _{u}+b\partial _{v}}$, with a and b smooth functions. If ${\displaystyle X}$ is a vector field and ${\displaystyle g}$ is a smooth function, then ${\displaystyle Xg}$ is also a smooth function. The first order differential operator ${\displaystyle X}$ is a derivation, i.e. it satisfies the Leibniz rule ${\displaystyle X(gh)=(Xg)h+g(Xh).}$^[19]

For vector fields X and Y it is simple to check that the operator ${\displaystyle [X,Y]=XY-YX}$ is a derivation corresponding to a vector field. It is called the Lie bracket ${\displaystyle [X,Y]}$. It is skew-symmetric ${\displaystyle [X,Y]=-[Y,X]}$ and satisfies the Jacobi identity:

${\displaystyle [[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0.}$

In summary, vector fields on ${\displaystyle U}$ or ${\displaystyle V}$ form a Lie algebra under the Lie bracket.^[20]

First and second fundamental forms, the shape operator, and the curvature

Let S be a regular surface in ℝ³. Given a local parametrization f : V → S and a unit normal vector field n to f(V), one defines the following objects as real-valued or matrix-valued functions on V. The first fundamental form depends only on f, and not on n. The fourth column records the way in which these functions depend on f, by relating the functions E ′, F ′, G ′, L ′, etc., arising for a different choice of local parametrization, f ′ : V ′ → S, to those arising for f. Here A denotes the Jacobian matrix of f ^–1 ∘ f ′. The key relation in establishing the formulas of the fourth column is then

${\displaystyle {\begin{pmatrix}{\frac {\partial f'}{\partial u}}\\{\frac {\partial f'}{\partial v}}\end{pmatrix}}=A{\begin{pmatrix}{\frac {\partial f}{\partial u}}\\{\frac {\partial f}{\partial v}}\end{pmatrix}},}$

as follows by the chain rule.

Terminology	Notation	Definition	Dependence on local parametrization
First fundamental form	E	${\displaystyle E={\frac {\partial f}{\partial u}}\cdot {\frac {\partial f}{\partial u}}}$	${\displaystyle {\begin{pmatrix}E'&F'\\F'&G'\end{pmatrix}}=A{\begin{pmatrix}E&F\\F&G\end{pmatrix}}A^{T}}$
	F	${\displaystyle F={\frac {\partial f}{\partial u}}\cdot {\frac {\partial f}{\partial v}}}$
	G	${\displaystyle G={\frac {\partial f}{\partial v}}\cdot {\frac {\partial f}{\partial v}}}$
Second fundamental form	L	${\displaystyle L={\frac {\partial ^{2}f}{\partial u^{2}}}\cdot n}$	${\displaystyle {\begin{pmatrix}L'&M'\\M'&N'\end{pmatrix}}=A{\begin{pmatrix}L&M\\M&N\end{pmatrix}}A^{T}}$
	M	${\displaystyle M={\frac {\partial ^{2}f}{\partial u\partial v}}\cdot n}$
	N	${\displaystyle N={\frac {\partial ^{2}f}{\partial v^{2}}}\cdot n}$
Shape operator[21]	P	${\displaystyle P={\begin{pmatrix}L&M\\M&N\end{pmatrix}}{\begin{pmatrix}E&F\\F&G\end{pmatrix}}^{-1}}$	${\displaystyle P'=APA^{-1}}$
Gaussian curvature	K	${\displaystyle K={\frac {LN-M^{2}}{EG-F^{2}}}}$	${\displaystyle K'=K}$
Mean curvature	H	${\displaystyle H={\frac {GL-2FM+EN}{2(EG-F^{2})}}}$	${\displaystyle H'=H}$
Principal curvatures	${\displaystyle \kappa _{\pm }}$	${\displaystyle H\pm {\sqrt {H^{2}-K}}}$	${\displaystyle \kappa _{\pm }'=\kappa _{\pm }}$

By a direct calculation with the matrix defining the shape operator, it can be checked that the Gaussian curvature is the determinant of the shape operator, the mean curvature is half of the trace of the shape operator, and the principal curvatures are the eigenvalues of the shape operator; moreover the Gaussian curvature is the product of the principal curvatures and the mean curvature is their sum. These observations can also be formulated as definitions of these objects. These observations also make clear that the last three rows of the fourth column follow immediately from the previous row, as similar matrices have identical determinant, trace, and eigenvalues. It is fundamental to note E, G, and EG − F² are all necessarily positive. This ensures that the matrix inverse in the definition of the shape operator is well-defined, and that the principal curvatures are real numbers.

Note also that a negation of the choice of unit normal vector field will negate the second fundamental form, the shape operator, the mean curvature, and the principal curvatures, but will leave the Gaussian curvature unchanged. In summary, this has shown that, given a regular surface S, the Gaussian curvature of S can be regarded as a real-valued function on S; relative to a choice of unit normal vector field on all of S, the two principal curvatures and the mean curvature are also real-valued functions on S.

Definition of second fundamental form

The principal curvatures at a point on a surface

Geometrically, the first and second fundamental forms can be viewed as giving information on how f(u, v) moves around in ℝ³ as (u, v) moves around in V. In particular, the first fundamental form encodes how quickly f moves, while the second fundamental form encodes the extent to which its motion is in the direction of the normal vector n. In other words, the second fundamental form at a point p encodes the length of the orthogonal projection from S to the tangent plane to S at p; in particular it gives the quadratic function which best approximates this length. This thinking can be made precise by the formulas

${\displaystyle {\begin{aligned}\lim _{(h,k)\to (0,0)}{\frac {{\big |}f(u+h,v+k)-f(u,v){\big |}^{2}-{\big (}Eh^{2}+2Fhk+Gk^{2}{\big )}}{h^{2}+k^{2}}}&=0\\\lim _{(h,k)\to (0,0)}{\frac {{\big (}f(u+h,v+k)-f(u,v){\big )}\cdot n-{\frac {1}{2}}{\big (}Lh^{2}+2Mhk+Nk^{2}{\big )}}{h^{2}+k^{2}}}&=0,\end{aligned}}}$

as follows directly from the definitions of the fundamental forms and Taylor's theorem in two dimensions. The principal curvatures can be viewed in the following way. At a given point p of S, consider the collection of all planes which contain the orthogonal line to S. Each such plane has a curve of intersection with S, which can be regarded as a plane curve inside of the plane itself. The two principal curvatures at p are the maximum and minimum possible values of the curvature of this plane curve at p, as the plane under consideration rotates around the normal line.

The following summarizes the calculation of the above quantities relative to a Monge patch f(u, v) = (u, v, h(u, v)). Here h_u and h_v denote the two partial derivatives of h, with analogous notation for the second partial derivatives. The second fundamental form and all subsequent quantities are calculated relative to the given choice of unit normal vector field.

Quantity	Formula
A unit normal vector field	${\displaystyle n={\frac {(-h_{u},-h_{v},1)}{\sqrt {1+h_{u}^{2}+h_{v}^{2}}}}}$
First fundamental form	${\displaystyle {\begin{pmatrix}E&F\\F&G\end{pmatrix}}={\begin{pmatrix}1+h_{u}^{2}&h_{u}h_{v}\\h_{u}h_{v}&1+h_{v}^{2}\end{pmatrix}}}$
Second fundamental form	${\displaystyle {\begin{pmatrix}L&M\\M&N\end{pmatrix}}={\frac {1}{\sqrt {1+h_{u}^{2}+h_{v}^{2}}}}{\begin{pmatrix}h_{uu}&h_{uv}\\h_{uv}&h_{vv}\end{pmatrix}}}$
Shape operator	${\displaystyle P={\frac {1}{(1+h_{u}^{2}+h_{v}^{2})^{3/2}}}{\begin{pmatrix}h_{uu}(1+h_{v}^{2})-h_{uv}h_{u}h_{v}&h_{uv}(1+h_{u}^{2})-h_{uu}h_{u}h_{v}\\h_{uv}(1+h_{v}^{2})-h_{vv}h_{u}h_{v}&h_{vv}(1+h_{u}^{2})-h_{uv}h_{u}h_{v}\end{pmatrix}}}$
Gaussian curvature	${\displaystyle K={\frac {h_{uu}h_{vv}-h_{uv}^{2}}{(1+h_{u}^{2}+h_{v}^{2})^{2}}}}$
Mean curvature	${\displaystyle H={\frac {(1+h_{v}^{2})h_{uu}-2h_{u}h_{v}h_{uv}+(1+h_{u}^{2})h_{vv}}{2(1+h_{u}^{2}+h_{v}^{2})^{3/2}}}}$

Christoffel symbols, Gauss–Codazzi equations, and the Theorema Egregium

Let S be a regular surface in ℝ³. The Christoffel symbols assign, to each local parametrization f : V → S, eight functions on V, defined by^[22]

${\displaystyle {\begin{pmatrix}\Gamma _{11}^{1}&\Gamma _{12}^{1}&\Gamma _{21}^{1}&\Gamma _{22}^{1}\\\Gamma _{11}^{2}&\Gamma _{12}^{2}&\Gamma _{21}^{2}&\Gamma _{22}^{2}\end{pmatrix}}={\begin{pmatrix}E&F\\F&G\end{pmatrix}}^{-1}{\begin{pmatrix}{\frac {1}{2}}{\frac {\partial E}{\partial u}}&{\frac {1}{2}}{\frac {\partial E}{\partial v}}&{\frac {1}{2}}{\frac {\partial E}{\partial v}}&{\frac {\partial F}{\partial v}}-{\frac {1}{2}}{\frac {\partial G}{\partial u}}\\{\frac {\partial F}{\partial u}}-{\frac {1}{2}}{\frac {\partial E}{\partial v}}&{\frac {1}{2}}{\frac {\partial G}{\partial u}}&{\frac {1}{2}}{\frac {\partial G}{\partial u}}&{\frac {1}{2}}{\frac {\partial G}{\partial v}}\end{pmatrix}}.}$

They can also be defined by the following formulas, in which n is a unit normal vector field along f(V) and L, M, N are the corresponding components of the second fundamental form:

${\displaystyle {\begin{aligned}{\frac {\partial ^{2}f}{\partial u^{2}}}&=\Gamma _{11}^{1}{\frac {\partial f}{\partial u}}+\Gamma _{11}^{2}{\frac {\partial f}{\partial v}}+Ln\\{\frac {\partial ^{2}f}{\partial u\partial v}}&=\Gamma _{12}^{1}{\frac {\partial f}{\partial u}}+\Gamma _{12}^{2}{\frac {\partial f}{\partial v}}+Mn\\{\frac {\partial ^{2}f}{\partial v^{2}}}&=\Gamma _{22}^{1}{\frac {\partial f}{\partial u}}+\Gamma _{22}^{2}{\frac {\partial f}{\partial v}}+Nn.\end{aligned}}}$

The key to this definition is that ∂f/∂u, ∂f/∂v, and n form a basis of ℝ³ at each point, relative to which each of the three equations uniquely specifies the Christoffel symbols as coordinates of the second partial derivatives of f. The choice of unit normal has no effect on the Christoffel symbols, since if n is exchanged for its negation, then the components of the second fundamental form are also negated, and so the signs of Ln, Mn, Nn are left unchanged.

The second definition shows, in the context of local parametrizations, that the Christoffel symbols are geometrically natural. Although the formulas in the first definition appear less natural, they have the importance of showing that the Christoffel symbols can be calculated from the first fundamental form, which is not immediately apparent from the second definition. The equivalence of the definitions can be checked by directly substituting the first definition into the second, and using the definitions of E, F, G.

The Codazzi equations assert that^[23]

${\displaystyle {\begin{aligned}{\frac {\partial L}{\partial v}}-{\frac {\partial M}{\partial u}}&=L\Gamma _{12}^{1}+M(\Gamma _{12}^{2}-\Gamma _{11}^{1})-N\Gamma _{11}^{2}\\{\frac {\partial M}{\partial v}}-{\frac {\partial N}{\partial u}}&=L\Gamma _{22}^{1}+M(\Gamma _{22}^{2}-\Gamma _{12}^{1})-N\Gamma _{12}^{2}.\end{aligned}}}$

These equations can be directly derived from the second definition of Christoffel symbols given above; for instance, the first Codazzi equation is obtained by differentiating the first equation with respect to v, the second equation with respect to u, subtracting the two, and taking the dot product with n. The Gauss equation asserts that^[24]

${\displaystyle {\begin{aligned}KE&={\frac {\partial \Gamma _{11}^{2}}{\partial v}}-{\frac {\partial \Gamma _{21}^{2}}{\partial u}}+\Gamma _{21}^{2}\Gamma _{11}^{1}+\Gamma _{22}^{2}\Gamma _{11}^{2}-\Gamma _{11}^{2}\Gamma _{21}^{1}-\Gamma _{12}^{2}\Gamma _{21}^{2}\\KF&={\frac {\partial \Gamma _{12}^{2}}{\partial v}}-{\frac {\partial \Gamma _{22}^{2}}{\partial u}}+\Gamma _{21}^{2}\Gamma _{12}^{1}-\Gamma _{11}^{2}\Gamma _{22}^{1}\\KG&={\frac {\partial \Gamma _{22}^{1}}{\partial u}}-{\frac {\partial \Gamma _{12}^{1}}{\partial v}}+\Gamma _{11}^{1}\Gamma _{22}^{1}+\Gamma _{12}^{1}\Gamma _{22}^{2}-\Gamma _{21}^{1}\Gamma _{12}^{1}-\Gamma _{22}^{1}\Gamma _{12}^{2}\end{aligned}}}$

These can be similarly derived as the Codazzi equations, with one using the Weingarten equations instead of taking the dot product with n. Although these are written as three separate equations, they are identical when the definitions of the Christoffel symbols, in terms of the first fundamental form, are substituted in. There are many ways to write the resulting expression, one of them derived in 1852 by Brioschi using a skillful use of determinants:^[25][26]

${\displaystyle K={\frac {1}{(EG-F^{2})^{2}}}\det {\begin{pmatrix}-{1 \over 2}{\frac {\partial ^{2}E}{\partial v^{2}}}+{\frac {\partial ^{2}F}{\partial u\partial v}}-{1 \over 2}{\frac {\partial ^{2}G}{\partial u^{2}}}&{1 \over 2}{\frac {\partial E}{\partial u}}&{\frac {\partial F}{\partial u}}-{1 \over 2}{\frac {\partial E}{\partial v}}\\{\frac {\partial F}{\partial v}}-{1 \over 2}{\frac {\partial G}{\partial u}}&E&F\\{1 \over 2}{\frac {\partial G}{\partial v}}&F&G\end{pmatrix}}-{\frac {1}{(EG-F^{2})^{2}}}\det {\begin{pmatrix}0&{1 \over 2}{\frac {\partial E}{\partial v}}&{1 \over 2}{\frac {\partial G}{\partial u}}\\{1 \over 2}{\frac {\partial E}{\partial v}}&E&F\\{1 \over 2}{\frac {\partial G}{\partial u}}&F&G\end{pmatrix}}.}$

When the Christoffel symbols are considered as being defined by the first fundamental form, the Gauss and Codazzi equations represent certain constraints between the first and second fundamental forms. The Gauss equation is particularly noteworthy, as it shows that the Gaussian curvature can be computed directly from the first fundamental form, without the need for any other information; equivalently, this says that LN − M² can actually be written as a function of E, F, G, even though the individual components L, M, N cannot. This is known as the theorema egregium, and was a major discovery of Carl Friedrich Gauss. It is particularly striking when one recalls the geometric definition of the Gaussian curvature of S as being defined by the maximum and minimum radii of osculating circles; they seem to be fundamentally defined by the geometry of how S bends within ℝ³. Nevertheless, the theorem shows that their product can be determined from the "intrinsic" geometry of S, having only to do with the lengths of curves along S and the angles formed at their intersections. As said by Marcel Berger:^[27]

This theorem is baffling. [...] It is the kind of theorem which could have waited dozens of years more before being discovered by another mathematician since, unlike so much of intellectual history, it was absolutely not in the air. [...] To our knowledge there is no simple geometric proof of the theorema egregium today.

The Gauss-Codazzi equations can also be succinctly expressed and derived in the language of connection forms due to Élie Cartan.^[28] In the language of tensor calculus, making use of natural metrics and connections on tensor bundles, the Gauss equation can be written as H² − |h|² = R and the two Codazzi equations can be written as ∇₁ h₁₂ = ∇₂ h₁₁ and ∇₁ h₂₂ = ∇₂ h₁₂; the complicated expressions to do with Christoffel symbols and the first fundamental form are completely absorbed into the definitions of the covariant tensor derivative ∇h and the scalar curvature R. Pierre Bonnet proved that two quadratic forms satisfying the Gauss-Codazzi equations always uniquely determine an embedded surface locally.^[29] For this reason the Gauss-Codazzi equations are often called the fundamental equations for embedded surfaces, precisely identifying where the intrinsic and extrinsic curvatures come from. They admit generalizations to surfaces embedded in more general Riemannian manifolds.

Isometries

A diffeomorphism ${\displaystyle \varphi }$ between open sets ${\displaystyle U}$ and ${\displaystyle V}$ in a regular surface ${\displaystyle S}$ is said to be an isometry if it preserves the metric, i.e. the first fundamental form.^[30][31][32] Thus for every point ${\displaystyle p}$ in ${\displaystyle U}$ and tangent vectors ${\displaystyle w_{1},\,\,w_{2}}$ at ${\displaystyle p}$, there are equalities

${\displaystyle E(p)w_{1}\cdot w_{1}+2F(p)w_{1}\cdot w_{2}+G(p)w_{2}\cdot w_{2}=E(\varphi (p))\varphi ^{\prime }(w_{1})\cdot \varphi ^{\prime }(w_{1})+2F(\varphi (p))\varphi ^{\prime }(w_{1})\cdot \varphi ^{\prime }(w_{2})+G(\varphi (p))\varphi ^{\prime }(w_{1})\cdot \varphi ^{\prime }(w_{2}).}$

In terms of the inner product coming from the first fundamental form, this can be rewritten as

${\displaystyle (w_{1},w_{2})_{p}=(\varphi ^{\prime }(w_{1}),\varphi ^{\prime }(w_{2}))_{\varphi (p)}}$.

The catenoid is a regular surface of revolution

On the other hand, the length of a parametrized curve ${\displaystyle \gamma (t)=(x(t),y(t))}$ can be calculated as

${\displaystyle L(\gamma )=\int _{a}^{b}{\sqrt {E{\dot {x}}\cdot {\dot {x}}+2F{\dot {x}}\cdot {\dot {y}}+G{\dot {y}}\cdot {\dot {y}}}}\,dt}$

and, if the curve lies in ${\displaystyle U}$, the rules for change of variables show that

${\displaystyle L(\varphi \circ \gamma )=L(\gamma ).}$

Conversely if ${\displaystyle \varphi }$ preserves the lengths of all parametrized in curves then ${\displaystyle \varphi }$ is an isometry. Indeed, for suitable choices of ${\displaystyle \gamma }$, the tangent vectors ${\displaystyle {\dot {x}}}$ and ${\displaystyle {\dot {y}}}$ give arbitrary tangent vectors ${\displaystyle w_{1}}$ and ${\displaystyle w_{2}}$. The equalities must hold for all choice of tangent vectors ${\displaystyle w_{1}}$ and ${\displaystyle w_{2}}$ as well as ${\displaystyle \varphi ^{\prime }(w_{1})}$ and ${\displaystyle \varphi ^{\prime }(w_{2})}$, so that ${\displaystyle (\varphi ^{\prime }(w_{1}),\varphi ^{\prime }(w_{2}))_{\varphi (p)}=(w_{1},w_{1})_{p}}$.^[33]

A simple example of an isometry is provided by two parametrizations ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ of an open set ${\displaystyle U}$ into regular surfaces ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$. If ${\displaystyle E_{1}=E_{2}}$, ${\displaystyle F_{1}=F_{2}}$ and ${\displaystyle G_{1}=G_{2}}$, then ${\displaystyle \varphi =f_{2}\circ f_{1}^{-1}}$ is an isometry of ${\displaystyle f_{1}(U)}$ onto ${\displaystyle f_{2}(U)}$. ^[34]

The cylinder and the plane give examples of surfaces that are locally isometric but which cannot be extended to an isometry for topological reasons.^[35] As another example, the catenoid and helicoid are locally isometric.^[36]

Covariant derivatives

A tangential vector field X on S assigns, to each p in S, a tangent vector X_p to S at p. According to the "intrinsic" definition of tangent vectors given above, a tangential vector field X then assigns, to each local parametrization f : V → S, two real-valued functions X¹ and X² on V, so that

${\displaystyle X_{p}=X^{1}{\big (}f^{-1}(p){\big )}{\frac {\partial f}{\partial u}}{\Big |}_{f^{-1}(p)}+X^{2}{\big (}f^{-1}(p){\big )}{\frac {\partial f}{\partial v}}{\Big |}_{f^{-1}(p)}}$

for each p in S. One says that X is smooth if the functions X¹ and X² are smooth, for any choice of f.^[37] According to the other definitions of tangent vectors given above, one may also regard a tangential vector field X on S as a map X : S → ℝ³ such that X(p) is contained in the tangent space T_pS ⊂ ℝ³ for each p in S. As is common in the more general situation of smooth manifolds, tangential vector fields can also be defined as certain differential operators on the space of smooth functions on S.

The covariant derivatives (also called "tangential derivatives") of Tullio Levi-Civita and Gregorio Ricci-Curbastro provide a means of differentiating smooth tangential vector fields. Given a tangential vector field X and a tangent vector Y to S at p, the covariant derivative ∇_YX is a certain tangent vector to S at p. Consequently, if X and Y are both tangential vector fields, then ∇_YX can also be regarded as a tangential vector field; iteratively, if X, Y, and Z are tangential vector fields, the one may compute ∇_Z∇_YX, which will be another tangential vector field. There are a few ways to define the covariant derivative; the first below uses the Christoffel symbols and the "intrinsic" definition of tangent vectors, and the second is more manifestly geometric.

Given a tangential vector field X and a tangent vector Y to S at p, one defines ∇_YX to be the tangent vector to p which assigns to a local parametrization f : V → S the two numbers

${\displaystyle (\nabla _{Y}X)^{k}=D_{(Y^{1},Y^{2})}X^{k}{\Big |}_{f^{-1}(p)}+\sum _{i=1}^{2}\sum _{j=1}^{2}{\big (}\Gamma _{ij}^{k}X^{j}{\big )}{\Big |}_{f^{-1}(p)}Y^{i},\qquad (k=1,2)}$

where D_{(Y¹, Y²)} is the directional derivative.^[38] This is often abbreviated in the less cumbersome form (∇_YX)^k = ∂_Y(X ^k) + Y ⁱΓ^k
_ijX ^j, making use of Einstein notation and with the locations of function evaluation being implicitly understood. This follows a standard prescription in Riemannian geometry for obtaining a connection from a Riemannian metric. It is a fundamental fact that the vector

${\displaystyle (\nabla _{Y}X)^{1}{\frac {\partial f}{\partial u}}+(\nabla _{Y}X)^{2}{\frac {\partial f}{\partial v}}}$

in ℝ³ is independent of the choice of local parametization f, although this is rather tedious to check.

One can also define the covariant derivative by the following geometric approach, which does not make use of Christoffel symbols or local parametrizations.^[39][40][41] Let X be a vector field on S, viewed as a function S → ℝ³. Given any curve c : (a, b) → S, one may consider the composition X ∘ c : (a, b) → ℝ³. As a map between Euclidean spaces, it can be differentiated at any input value to get an element (X ∘ c)′(t) of ℝ³. The orthogonal projection of this vector onto T_c(t)S defines the covariant derivative ∇_{c ′(t)}X. Although this is a very geometrically clean definition, it is necessary to show that the result only depends on c′(t) and X, and not on c and X; local parametrizations can be used for this small technical argument.

It is not immediately apparent from the second definition that covariant differentiation depends only on the first fundamental form of S; however, this is immediate from the first definition, since the Christoffel symbols can be defined directly from the first fundamental form. It is straightforward to check that the two definitions are equivalent. The key is that when one regards X¹∂f/∂u + X²∂f/∂v as a ℝ³-valued function, its differentiation along a curve results in second partial derivatives ∂²f; the Christoffel symbols enter with orthogonal projection to the tangent space, due to the formulation of the Christoffel symbols as the tangential components of the second derivatives of f relative to the basis ∂f/∂u, ∂f/∂v, n.^[38] This is discussed in the above section.

The right-hand side of the three Gauss equations can be expressed using covariant differentiation. For instance, the right-hand side

${\displaystyle {\frac {\partial \Gamma _{11}^{2}}{\partial v}}-{\frac {\partial \Gamma _{21}^{2}}{\partial u}}+\Gamma _{21}^{2}\Gamma _{11}^{1}+\Gamma _{22}^{2}\Gamma _{11}^{2}-\Gamma _{11}^{2}\Gamma _{21}^{1}-\Gamma _{12}^{2}\Gamma _{21}^{2}}$

can be recognized as the second coordinate of

${\displaystyle \nabla _{\frac {\partial f}{\partial v}}\nabla _{\frac {\partial f}{\partial u}}{\frac {\partial f}{\partial u}}-\nabla _{\frac {\partial f}{\partial u}}\nabla _{\frac {\partial f}{\partial v}}{\frac {\partial f}{\partial u}}}$

relative to the basis ∂f/∂u, ∂f/∂v, as can be directly verified using the definition of covariant differentiation by Christoffel symbols. In the language of Riemannian geometry, this observation can also be phrased as saying that the right-hand sides of the Gauss equations are various components of the Ricci curvature of the Levi-Civita connection of the first fundamental form, when interpreted as a Riemannian metric.

Surfaces of revolution

A surface of revolution is obtained by rotating a curve in the xz-plane about the z-axis. Such surfaces include spheres, cylinders, cones, tori, and the catenoid. The general ellipsoids, hyperboloids, and paraboloids are not. Suppose that the curve is parametrized by

${\displaystyle x=c_{1}(s),\,\,z=c_{2}(s)}$

with s drawn from an interval (a, b). If c₁ is never zero, if c₁′ and c₂′ are never both equal to zero, and if c₁ and c₂ are both smooth, then the corresponding surface of revolution

${\displaystyle S={\Big \{}{\big (}c_{1}(s)\cos t,c_{1}(s)\sin t,c_{2}(s){\big )}\colon s\in (a,b){\text{ and }}t\in \mathbb {R} {\Big \}}}$

will be a regular surface in ℝ³. A local parametrization f : (a, b) × (0, 2π) → S is given by

${\displaystyle f(s,t)={\big (}c_{1}(s)\cos t,c_{1}(s)\sin t,c_{2}(s){\big )}.}$

Relative to this parametrization, the geometric data is:^[42]

Quantity	Formula
A unit normal vector field	${\displaystyle n={\frac {{\big (}-c_{2}'(s)\cos t,-c_{2}'(s)\sin t,c_{1}'(s){\big )}}{\sqrt {c_{1}'(s)^{2}+c_{2}'(s)^{2}}}}}$
First fundamental form	${\displaystyle {\begin{pmatrix}E&F\\F&G\end{pmatrix}}={\begin{pmatrix}c_{1}'(s)^{2}+c_{2}'(s)^{2}&0\\0&c_{1}(s)^{2}\end{pmatrix}}}$
Second fundamental form	${\displaystyle {\begin{pmatrix}L&M\\M&N\end{pmatrix}}={\begin{pmatrix}{\frac {c_{1}'(s)c_{2}''(s)-c_{2}'(s)c_{1}''(s)}{\sqrt {c_{1}'(s)^{2}+c_{2}'(s)^{2}}}}&0\\0&{\frac {c_{1}(s)c_{2}'(s)}{\sqrt {c_{1}'(s)^{2}+c_{2}'(s)^{2}}}}\end{pmatrix}}}$
Principal curvatures	${\displaystyle {\frac {c_{1}'(s)c_{2}''(s)-c_{2}'(s)c_{1}''(s)}{{\big (}c_{1}'(s)^{2}+c_{2}'(s)^{2}{\big )}^{3/2}}}\quad {\text{and}}\quad {\frac {c_{2}'(s)}{c_{1}(s){\sqrt {c_{1}'(s)^{2}+c_{2}'(s)^{2}}}}}}$
Gaussian curvature	${\displaystyle K={\frac {c_{2}'(s){\big (}c_{1}'(s)c_{2}''(s)-c_{2}'(s)c_{1}''(s){\big )}}{c_{1}(s){\big (}c_{1}'(s)^{2}+c_{2}'(s)^{2}{\big )}^{2}}}}$
Mean curvature	${\displaystyle H={\frac {1}{2}}{\frac {c_{1}'(s)c_{2}''(s)-c_{2}'(s)c_{1}''(s)}{{\big (}c_{1}'(s)^{2}+c_{2}'(s)^{2}{\big )}^{3/2}}}+{\frac {1}{2}}{\frac {c_{2}'(s)}{c_{1}(s){\sqrt {c_{1}'(s)^{2}+c_{2}'(s)^{2}}}}}.}$

In the special case that the original curve is parametrized by arclength, i.e. (c₁′(s))² + (c₂′(s))² = 1, one can differentiate to find c₁′(s)c₁′′(s) + c₂′(s)c₂′′(s) = 0. On substitution into the Gaussian curvature, one has the simplified

${\displaystyle K=-{\frac {c_{1}''(s)}{c_{1}(s)}}\qquad {\text{and}}\qquad H=c_{1}'(s)c_{2}''(s)-c_{2}'(s)c_{1}''(s)+{\frac {c_{2}'(s)}{c_{1}(s)}}.}$

The simplicity of this formula makes it particularly easy to study the class of rotationally symmetric surfaces with constant Gaussian curvature.^[43] By reduction to the alternative case that c₂(s) = s, one can study the rotationally symmetric minimal surfaces, with the result that any such surface is part of a plane or a scaled catenoid.^[44]

Each constant-t curve on S can be parametrized as a geodesic; a constant-s curve on S can be parametrized as a geodesic if and only if c₁′(s) is equal to zero. Generally, geodesics on S are governed by Clairaut's relation.

A quadric ellipsoid

Quadric surfaces

Consider the quadric surface defined by^[45]

${\displaystyle {x^{2} \over a}+{y^{2} \over b}+{z^{2} \over c}=1.}$

This surface admits a parametrization

${\displaystyle x={\sqrt {a(a-u)(a-v) \over (a-b)(a-c)}},\,\,y={\sqrt {b(b-u)(b-v) \over (b-a)(b-c)}},\,\,z={\sqrt {c(c-u)(c-v) \over (c-b)(c-a)}}.}$

The Gaussian curvature and mean curvature are given by

${\displaystyle K={abc \over u^{2}v^{2}},\,\,K_{m}=-(u+v){\sqrt {abc \over u^{3}v^{3}}}.}$

A single-sheeted quadric hyperboloid which is a ruled surface in two different ways.

Ruled surfaces

A ruled surface is one which can be generated by the motion of a straight line in E³.^[46] Choosing a directrix on the surface, i.e. a smooth unit speed curve c(t) orthogonal to the straight lines, and then choosing u(t) to be unit vectors along the curve in the direction of the lines, the velocity vector v = c_t and u satisfy

${\displaystyle u\cdot v=0,\,\,\|u\|=1,\,\,\|v\|=1.}$

The surface consists of points

${\displaystyle c(t)+s\cdot u(t)}$

as s and t vary.

Then, if

${\displaystyle a=\|u_{t}\|,\,\,b=u_{t}\cdot v,\,\,\alpha =-{\frac {b}{a^{2}}},\,\,\beta ={\frac {\sqrt {a^{2}-b^{2}}}{a^{2}}},}$

the Gaussian and mean curvature are given by

${\displaystyle K=-{\beta ^{2} \over ((s-\alpha )^{2}+\beta ^{2})^{2}},\,\,K_{m}=-{r[(s-\alpha )^{2}+\beta ^{2})]+\beta _{t}(s-\alpha )+\beta \alpha _{t} \over [(s-\alpha )^{2}+\beta ^{2}]^{\frac {3}{2}}}.}$

The Gaussian curvature of the ruled surface vanishes if and only if u_t and v are proportional,^[47] This condition is equivalent to the surface being the envelope of the planes along the curve containing the tangent vector v and the orthogonal vector u, i.e. to the surface being developable along the curve.^[48] More generally a surface in E³ has vanishing Gaussian curvature near a point if and only if it is developable near that point.^[49] (An equivalent condition is given below in terms of the metric.)

Minimal surfaces

In 1760 Lagrange extended Euler's results on the calculus of variations involving integrals in one variable to two variables.^[50] He had in mind the following problem:

Given a closed curve in E³, find a surface having the curve as boundary with minimal area.

Such a surface is called a minimal surface.

In 1776 Jean Baptiste Meusnier showed that the differential equation derived by Lagrange was equivalent to the vanishing of the mean curvature of the surface:

A surface is minimal if and only if its mean curvature vanishes.

Minimal surfaces have a simple interpretation in real life: they are the shape a soap film will assume if a wire frame shaped like the curve is dipped into a soap solution and then carefully lifted out. The question as to whether a minimal surface with given boundary exists is called Plateau's problem after the Belgian physicist Joseph Plateau who carried out experiments on soap films in the mid-nineteenth century. In 1930 Jesse Douglas and Tibor Radó gave an affirmative answer to Plateau's problem (Douglas was awarded one of the first Fields medals for this work in 1936).^[51]

Many explicit examples of minimal surface are known explicitly, such as the catenoid, the helicoid, the Scherk surface and the Enneper surface. There has been extensive research in this area, summarised in Osserman (2002). In particular a result of Osserman shows that if a minimal surface is non-planar, then its image under the Gauss map is dense in S².

Surfaces with (from l. to r.) constant negative, zero and positive Gaussian curvature

Surfaces of constant Gaussian curvature

Eugenio Beltrami (1835-1899)

If a surface has constant Gaussian curvature, it is called a surface of constant curvature.^[52]

• The unit sphere in E³ has constant Gaussian curvature +1.
• The Euclidean plane and the cylinder both have constant Gaussian curvature 0.
• The surfaces of revolution with φ_tt = φ have constant Gaussian curvature –1. Particular cases are obtained by taking φ(t) =C cosh t, C sinh t and C e^t.^[53] The latter case is the classical pseudosphere generated by rotating a tractrix around a central axis. In 1868 Eugenio Beltrami showed that the geometry of the pseudosphere was directly related to that of the hyperbolic plane, discovered independently by Lobachevsky (1830) and Bolyai (1832). Already in 1840, F. Minding, a student of Gauss, had obtained trigonometric formulas for the pseudosphere identical to those for the hyperbolic plane.^[54] The intrinsic geometry of this surface is now better understood in terms of the Poincaré metric on the upper half plane or the unit disc, and has been described by other models such as the Klein model or the hyperboloid model, obtained by considering the two-sheeted hyperboloid q(x, y, z) = −1 in three-dimensional Minkowski space, where q(x, y, z) = x² + y² – z².^[55]

Each of these surfaces of constant curvature has a transitive Lie group of symmetries. This group theoretic fact has far-reaching consequences, all the more remarkable because of the central role these special surfaces play in the geometry of surfaces, due to Poincaré's uniformization theorem (see below).

Other examples of surfaces with Gaussian curvature 0 include cones, tangent developables, and more generally any developable surface.

Shape operator

Wilhelm Blaschke (1885-1962)

The differential dn of the Gauss map n can be used to define a type of extrinsic curvature, known as the shape operator^[56] or Weingarten map. This operator first appeared implicitly in the work of Wilhelm Blaschke and later explicitly in a treatise by Burali-Forti and Burgati.^[57] Since at each point x of the surface, the tangent space is an inner product space, the shape operator S_x can be defined as a linear operator on this space by the formula

${\displaystyle (S_{x}v,w)=(dn(v),w)}$

for tangent vectors v, w (the inner product makes sense because dn(v) and w both lie in E³).^{[lower-alpha 1]} The right hand side is symmetric in v and w, so the shape operator is self-adjoint on the tangent space. The eigenvalues of S_x are just the principal curvatures k₁ and k₂ at x. In particular the determinant of the shape operator at a point is the Gaussian curvature, but it also contains other information, since the mean curvature is half the trace of the shape operator. The mean curvature is an extrinsic invariant. In intrinsic geometry, a cylinder is developable, meaning that every piece of it is intrinsically indistinguishable from a piece of a plane since its Gauss curvature vanishes identically. Its mean curvature is not zero, though; hence extrinsically it is different from a plane.

Equivalently, the shape operator can be defined as a linear operator on tangent spaces, S_p: T_pM→T_pM. If n is a unit normal field to M and v is a tangent vector then

${\displaystyle S(v)=\pm \nabla _{v}n}$

(there is no standard agreement whether to use + or − in the definition).

In general, the eigenvectors and eigenvalues of the shape operator at each point determine the directions in which the surface bends at each point. The eigenvalues correspond to the principal curvatures of the surface and the eigenvectors are the corresponding principal directions. The principal directions specify the directions that a curve embedded in the surface must travel to have maximum and minimum curvature, these being given by the principal curvatures.

Geodesics

A geodesic triangle on the sphere. The geodesics are great circle arcs.

Given a piecewise smooth path c(t) = (x(t), y(t)) in the chart for t in [a, b], its length is defined by

${\displaystyle L(c)=\int _{a}^{b}(E{\dot {x}}^{2}+2F{\dot {x}}{\dot {y}}+G{\dot {y}}^{2})^{\frac {1}{2}}\,dt}$

and energy by

${\displaystyle E(c)=\int _{a}^{b}(E{\dot {x}}^{2}+2F{\dot {x}}{\dot {y}}+G{\dot {y}}^{2})\,dt.}$

The length is independent of the parametrization of a path. By the Euler–Lagrange equations, if c(t) is a path minimising length, parametrized by arclength, it must satisfy the Euler equations

${\displaystyle {\ddot {x}}+\Gamma _{11}^{1}{\dot {x}}^{2}+2\Gamma _{12}^{1}{\dot {x}}{\dot {y}}+\Gamma _{22}^{1}{\dot {y}}^{2}=0}$

${\displaystyle {\ddot {y}}+\Gamma _{11}^{2}{\dot {x}}^{2}+2\Gamma _{12}^{2}{\dot {x}}{\dot {y}}+\Gamma _{22}^{2}{\dot {y}}^{2}=0}$

where the Christoffel symbols Γ^k
_ij are given by

${\displaystyle \Gamma _{ij}^{k}={\tfrac {1}{2}}\sum _{m}g^{km}(\partial _{j}g_{im}+\partial _{i}g_{jm}-\partial _{m}g_{ij})}$

where g₁₁ = E, g₁₂ = F, g₂₂ = G and g^ij is the inverse matrix to g_ij. A path satisfying the Euler equations is called a geodesic. By the Cauchy–Schwarz inequality a path minimising energy is just a geodesic parametrised by arc length; and, for any geodesic, the parameter t is proportional to arclength.^[58]

Geodesic curvature

The geodesic curvature k_g at a point of a curve c(t), parametrised by arc length, on an oriented surface is defined to be^[59]

${\displaystyle k_{g}={\ddot {c}}(t)\cdot \mathbf {n} (t).}$

where n(t) is the "principal" unit normal to the curve in the surface, constructed by rotating the unit tangent vector ċ(t) through an angle of +90°.

• The geodesic curvature at a point is an intrinsic invariant depending only on the metric near the point.
• A unit speed curve on a surface is a geodesic if and only if its geodesic curvature vanishes at all points on the curve.
• A unit speed curve c(t) in an embedded surface is a geodesic if and only if its acceleration vector c̈(t) is normal to the surface.

The geodesic curvature measures in a precise way how far a curve on the surface is from being a geodesic.

Orthogonal coordinates

When F = 0 throughout a coordinate chart, such as with the geodesic polar coordinates discussed below, the images of lines parallel to the x- and y-axes are orthogonal and provide orthogonal coordinates. If H = (EG)^1⁄2, then the Gaussian curvature is given by^[60]

${\displaystyle K=-{1 \over 2H}\left[\partial _{x}\left({\frac {G_{x}}{H}}\right)+\partial _{y}\left({\frac {E_{y}}{H}}\right)\right].}$

If in addition E = 1, so that H = G^1⁄2, then the angle φ at the intersection between geodesic (x(t),y(t)) and the line y = constant is given by the equation

${\displaystyle \tan \varphi =H\cdot {\frac {\dot {y}}{\dot {x}}}.}$

The derivative of φ is given by a classical derivative formula of Gauss:^[61]

${\displaystyle {\dot {\varphi }}=-H_{x}\cdot {\dot {y}}.}$

Exponential map

The theory of ordinary differential equations shows that if f(t, v) is smooth then the differential equation dv/dt = f(t, v) with initial condition v(0) = v₀ has a unique solution for |t| sufficiently small and the solution depends smoothly on t and v₀. This implies that for sufficiently small tangent vectors v at a given point p = (x₀, y₀), there is a geodesic c_v(t) defined on (−2, 2) with c_v(0) = (x₀, y₀) and ċ_v(0) = v. Moreover, if |s| ≤ 1, then c_sv = c_v(st). The exponential map is defined by

exp_p(v) = c_v (1)

and gives a diffeomorphism between a disc ‖v‖ < δ and a neighbourhood of p; more generally the map sending (p, v) to exp_p(v) gives a local diffeomorphism onto a neighbourhood of (p, p). The exponential map gives geodesic normal coordinates near p.^[64]