Inmathematics, thedifferential geometry of surfaces deals with thedifferential geometry ofsmoothsurfaces^[a] with various additional structures, most often, aRiemannian metric.^[b]

Surfaces have been extensively studied from various perspectives:extrinsically, relating to theirembedding inEuclidean space andintrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is theGaussian curvature, first studied in depth byCarl Friedrich Gauss,^[1] who showed that curvature was an intrinsic property of a surface, independent of itsisometric embedding in Euclidean space.

Surfaces naturally arise asgraphs offunctions of a pair ofvariables, and sometimes appear in parametric form or asloci associated tospace curves. An important role in their study has been played byLie groups (in the spirit of theErlangen program), namely thesymmetry groups of theEuclidean plane, thesphere and thehyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry throughconnections. On the other hand, extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. This is well illustrated by the non-linearEuler–Lagrange equations in thecalculus of variations: although Euler developed the one variable equations to understandgeodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was tominimal surfaces, a concept that can only be defined in terms of an embedding.

The volumes of certainquadric surfaces ofrevolution were calculated byArchimedes.^[2] The development ofcalculus in the seventeenth century provided a more systematic way of computing them.^[3] Curvature of general surfaces was first studied byEuler. In 1760^[4] he proved a formula for the curvature of a plane section of a surface and in 1771^[5] he considered surfaces represented in a parametric form.Monge laid down the foundations of their theory in his classical memoirL'application de l'analyse à la géometrie which appeared in 1795. The defining contribution to the theory of surfaces was made byGauss in two remarkable papers written in 1825 and 1827.^[1] This marked a new departure from tradition because for the first time Gauss considered theintrinsic geometry of a surface, the properties which are determined only by the geodesic distances between points on the surface independently of the particular way in which the surface is located in the ambient Euclidean space. The crowning result, theTheorema Egregium of Gauss, established that theGaussian curvature is an intrinsic invariant, i.e. invariant under localisometries. This point of view was extended to higher-dimensional spaces byRiemann and led to what is known today asRiemannian geometry. The nineteenth century was the golden age for the theory of surfaces, from both the topological and the differential-geometric point of view, with most leading geometers devoting themselves to their study.^{[citation needed]}Darboux collected many results in his four-volume treatiseThéorie des surfaces (1887–1896).

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It is intuitively quite familiar to say that the leaf of a plant, the surface of a glass, or the shape of a face, are curved in certain ways, and that all of these shapes, even after ignoring any distinguishing markings, have certain geometric features which distinguish one from another. The differential geometry of surfaces is concerned with a mathematical understanding of such phenomena. The study of this field, which was initiated in its modern form in the 1700s, has led to the development of higher-dimensional and abstract geometry, such asRiemannian geometry andgeneral relativity.^{[original research?]}

The essential mathematical object is that of aregular surface. Although conventions vary in their precise definition, these form a general class of subsets of three-dimensionalEuclidean space (ℝ³) which capture part of the familiar notion of "surface." By analyzing the class of curves which lie on such a surface, and the degree to which the surfaces force them to curve inℝ³, one can associate to each point of the surface two numbers, called theprincipal curvatures. Their average is called themean curvature of the surface, and their product is called theGaussian curvature.

There are many classic examples of regular surfaces, including:

• familiar examples such as planes, cylinders, and spheres
• minimal surfaces, which are defined by the property that their mean curvature is zero at every point. The best-known examples arecatenoids andhelicoids, although many more have been discovered. Minimal surfaces can also be defined by properties to do withsurface area, with the consequence that they provide a mathematical model for the shape ofsoap films when stretched across a wire frame
• ruled surfaces, which are surfaces that have at least one straight line running through every point; examples include the cylinder and thehyperboloid of one sheet.

A surprising result ofCarl Friedrich Gauss, known as theTheorema Egregium, showed that the Gaussian curvature of a surface, which by its definition has to do with how curves on the surface change directions in three dimensional space, can actually be measured by the lengths of curves lying on the surfaces together with the angles made when two curves on the surface intersect. Terminologically, this says that the Gaussian curvature can be calculated from thefirst fundamental form (also calledmetric tensor) of the surface. Thesecond fundamental form, by contrast, is an object which encodes how lengths and angles of curves on the surface are distorted when the curves are pushed off of the surface.

Despite measuring different aspects of length and angle, the first and second fundamental forms are not independent from one another, and they satisfy certain constraints called theGauss–Codazzi equations. A major theorem, often called the fundamental theorem of the differential geometry of surfaces, asserts that whenever two objects satisfy the Gauss-Codazzi constraints, they will arise as the first and second fundamental forms of a regular surface.

Using the first fundamental form, it is possible to define new objects on a regular surface.Geodesics are curves on the surface which satisfy a certain second-orderordinary differential equation which is specified by the first fundamental form. They are very directly connected to the study of lengths of curves; a geodesic of sufficiently short length will always be the curve of shortest length on the surface which connects its two endpoints. Thus, geodesics are fundamental to the optimization problem of determining the shortest path between two given points on a regular surface.

One can also defineparallel transport along any given curve, which gives a prescription for how to deform a tangent vector to the surface at one point of the curve to tangent vectors at all other points of the curve. The prescription is determined by a first-orderordinary differential equation which is specified by the first fundamental form.

The above concepts are essentially all to do with multivariable calculus. TheGauss–Bonnet theorem is a more global result, which relates the Gaussian curvature of a surface together with its topological type. It asserts that the average value of the Gaussian curvature is completely determined by theEuler characteristic of the surface together with its surface area.

Any regular surface is an example both of aRiemannian manifold andRiemann surface. Essentially all of the theory of regular surfaces as discussed here has a generalization in the theory of Riemannian manifolds and their submanifolds.

It has been suggested that this section besplit out into another article titledRegular surface (differential geometry). (Discuss)(February 2025)

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 betweenEuclidean 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]

A regular surface in Euclidean spaceℝ³ is a subsetS ofℝ³ such that every point ofS admits any of the following three concepts:local parametrizations,Monge patches, orimplicit functions.The following table gives definitions of such objects; Monge patches is perhaps the most visually intuitive, as it essentially says that a regular surface is a subset ofℝ³ which is locally the graph of a smooth function (whether over a region in theyz plane, thexz plane, or thexy plane).

The homeomorphisms appearing in the first definition are known aslocal parametrizations orlocal coordinate systems orlocal charts onS.^[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 asMonge patches. FunctionsF as in the third definition are calledlocal defining functions. The equivalence of all three definitions follows from theimplicit function theorem.^[14][15][16]

Given any two local parametrizationsf :V →U andf ′ :V ′→U ′ of a regular surface, the compositionf ⁻¹ ∘f ′ is necessarily smooth as a map between open subsets ofℝ².^[17] This shows that any regular surface naturally has the structure of asmooth 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∂_uf and∂_vf of a local parametrization may fail to belinearly independent. In this case,S may have singularities such ascuspidal edges. Such surfaces are typically studied insingularity theory. Other weakened forms of regular surfaces occur incomputer-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.^{[citation needed]}

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), takingh(u,v) = ± (1 −u² −v²)^1/2. It can also be covered by two local parametrizations, usingstereographic projection. The set{(x,y,z) : ((x² +y²)^1/2 −r)² +z² =R²} is atorus of revolution with radiir andR. It is a regular surface; local parametrizations can be given of the form

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

Thehyperboloid on two sheets{(x,y,z) :z² = 1 +x² +y²} is a regular surface; it can be covered by two Monge patches, withh(u,v) = ±(1 +u² +v²)^1/2. Thehelicoid appears in the theory ofminimal surfaces. It is covered by a single local parametrization,f(u,v) = (u sinv,u cosv,v).

LetS be a regular surface inℝ³, and letp be an element ofS. Using any of the above definitions, one can single out certain vectors inℝ³ as being tangent toS atp, and certain vectors inℝ³ as being orthogonal toS atp.

Objects used in definition	A vectorX inℝ3 is tangent toS atp if...	A vectorn inℝ3 is normal toS atp if...
Local parametrizations	... given any local parametrizationf :V →S withp ∈f(V),X is a linear combination of${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }u}{|}_{f^{-1}(p)}}$ and${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }v}{|}_{f^{-1}(p)}}$	... it is orthogonal to every tangent vector toS atp
Monge patches	... for any Monge patch(u,v) ↦ (u,v,h(u,v)) whose range includesp, one has ${\displaystyle X^1\frac{\mathrm{\partial }h}{\mathrm{\partial }u}+X^2\frac{\mathrm{\partial }h}{\mathrm{\partial }v}-X^3=0,}$ 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 includesp,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 functionF whose domain containsp,X is orthogonal to∇F(p)	... for any local defining functionF whose domain containsp,n is a multiple of∇F(p)

One sees that thetangent space ortangent plane toS atp, which is defined to consist of all tangent vectors toS atp, is a two-dimensional linear subspace ofℝ³; it is often denoted byT_pS. Thenormal space toS atp, which is defined to consist of all normal vectors toS atp, is a one-dimensional linear subspace ofℝ³ which is orthogonal to the tangent spaceT_pS. As such, at each pointp ofS, there are two normal vectors of unit length (unit normal vectors). The unit normal vectors atp can be given in terms of local parametrizations, Monge patches, or local defining functions, via the formulas

${\displaystyle \pm {\frac{\frac{\mathrm{\partial }f}{\mathrm{\partial }u}\times \frac{\mathrm{\partial }f}{\mathrm{\partial }v}}{\Vert \frac{\mathrm{\partial }f}{\mathrm{\partial }u}\times \frac{\mathrm{\partial }f}{\mathrm{\partial }v}\Vert }|}_{f^{-1}(p)},\pm {\frac{(\frac{\mathrm{\partial }h}{\mathrm{\partial }u},\frac{\mathrm{\partial }h}{\mathrm{\partial }v},-1)}{\sqrt{1+(\frac{\mathrm{\partial }h}{\mathrm{\partial }u}{)}^2+(\frac{\mathrm{\partial }h}{\mathrm{\partial }v}{)}^2}}|}_{(p_1,p_2)},\text{or}\pm \frac{\mathrm{\nabla }F(p)}{\Vert \mathrm{\nabla }F(p)\Vert },}$

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 ofsmooth manifolds. It defines the tangent space as an abstract two-dimensional real vector space, rather than as a linear subspace ofℝ³. In this definition, one says that a tangent vector toS atp is an assignment, to each local parametrizationf :V →S withp ∈f(V), of two numbersX¹ andX², such that for any other local parametrizationf ′ :V →S withp ∈f(V) (and with corresponding numbers(X ′)¹ and(X ′)²), one has

${\displaystyle \left(\begin{array}{c}{X}^1\\ X^2\end{array}\right)=A_{f^{\prime }(p)}\left(\begin{array}{c}(X^{\prime }{)}^1\\ (X^{\prime }{)}^2\end{array}\right),}$

whereA_{f ′(p)} is theJacobian matrix of the mappingf ⁻¹ ∘f ′, evaluated at the pointf ′(p). The collection of tangent vectors toS atp 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

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

inℝ³. The Jacobian condition onX¹ andX² ensures, by thechain rule, that this vector does not depend onf.

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${\displaystyle f:U\to S}$ a coordinate chart. If${\displaystyle V=f(U)}$, the space${\displaystyle C^{\mathrm{\infty }}(U)}$ can be identified with${\displaystyle C^{\mathrm{\infty }}(V)}$. Similarly${\displaystyle f}$ identifies vector fields on${\displaystyle U}$ with vector fields on${\displaystyle V}$. Taking standard variablesu andv, a vector field has the form${\displaystyle X=a{\mathrm{\partial }}_u+b{\mathrm{\partial }}_v}$, witha andb 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 aderivation, i.e. it satisfies the Leibniz rule${\displaystyle X(gh)=(Xg)h+g(Xh).}$^[19]

For vector fieldsX andY it is simple to check that the operator${\displaystyle [X,Y]=XY-YX}$ is a derivation corresponding to a vector field. It is called theLie 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 aLie algebra under the Lie bracket.^[20]

LetS be a regular surface inℝ³. Given a local parametrizationf :V →S and a unit normal vector fieldn tof(V), one defines the following objects as real-valued or matrix-valued functions onV. The first fundamental form depends only onf, and not onn. The fourth column records the way in which these functions depend onf, by relating the functionsE ′,F ′,G ′,L ′, etc., arising for a different choice of local parametrization,f ′ :V ′ →S, to those arising forf. HereA denotes theJacobian matrix off ^–1 ∘f ′. The key relation in establishing the formulas of the fourth column is then

${\displaystyle \left(\begin{array}{c}\frac{\mathrm{\partial }{f}^{\prime }}{\mathrm{\partial }u}\\ \frac{\mathrm{\partial }{f}^{\prime }}{\mathrm{\partial }v}\end{array}\right)=A\left(\begin{array}{c}\frac{\mathrm{\partial }f}{\mathrm{\partial }u}\\ \frac{\mathrm{\partial }f}{\mathrm{\partial }v}\end{array}\right),}$

as follows by thechain rule.

Terminology	Notation	Definition	Dependence on local parametrization
First fundamental form	E	${\displaystyle E=\frac{\mathrm{\partial }f}{\mathrm{\partial }u}\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }u}}$
	F	${\displaystyle F=\frac{\mathrm{\partial }f}{\mathrm{\partial }u}\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }v}}$
	G	${\displaystyle G=\frac{\mathrm{\partial }f}{\mathrm{\partial }v}\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }v}}$
Second fundamental form	L	${\displaystyle L=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{u}^2}\cdot n}$
	M	${\displaystyle M=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }u\mathrm{\partial }v}\cdot n}$
	N	${\displaystyle N=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{v}^2}\cdot n}$
Shape operator[21]	P		${\displaystyle P^{\prime }=AP{A}^{-1}}$
Gaussian curvature	K	${\displaystyle K=\frac{LN-M^2}{EG-F^2}}$	${\displaystyle K^{\prime }=K}$
Mean curvature	H	${\displaystyle H=\frac{GL-2FM+EN}{2(EG-F^2)}}$	${\displaystyle H^{\prime }=H}$
Principal curvatures	${\displaystyle {\kappa }_{\pm }}$	${\displaystyle H\pm \sqrt{H^2-K}}$	${\displaystyle {\kappa }_{\pm }^{\prime }={\kappa }_{\pm }}$

By a direct calculation with the matrix defining the shape operator, it can be checked that the Gaussian curvature is thedeterminant of the shape operator, the mean curvature is half of thetrace of the shape operator, and the principal curvatures are theeigenvalues 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, assimilar matrices have identical determinant, trace, and eigenvalues. It is fundamental to noteE,G, andEG −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 surfaceS, the Gaussian curvature ofS can be regarded as a real-valued function onS; relative to a choice of unit normal vector field on all ofS, the two principal curvatures and the mean curvature are also real-valued functions onS.

Geometrically, the first and second fundamental forms can be viewed as giving information on howf(u,v) moves around inℝ³ as(u,v) moves around inV. In particular, the first fundamental form encodes how quicklyf moves, while the second fundamental form encodes the extent to which its motion is in the direction of the normal vectorn. In other words, the second fundamental form at a pointp encodes the length of the orthogonal projection fromS to the tangent plane toS atp; in particular it gives the quadratic function which best approximates this length. This thinking can be made precise by the formulas

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

The following summarizes the calculation of the above quantities relative to a Monge patchf(u,v) = (u,v,h(u,v)). Hereh_u andh_v denote the two partial derivatives ofh, 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
Second fundamental form
Shape operator
Gaussian curvature	${\displaystyle K=\frac{h_{uu}{h}_{vv}-h_{uv}^2}{(1+h_u^2+h_v^2{)}^2}}$
Mean curvature	${\displaystyle H=\frac{(1+h_v^2)h_{uu}-2h_u{h}_v{h}_{uv}+(1+h_u^2)h_{vv}}{2(1+h_u^2+h_v^2{)}^{3/2}}}$

LetS be a regular surface inℝ³. TheChristoffel symbols assign, to each local parametrizationf :V →S, eight functions onV, defined by^[22]

They can also be defined by the following formulas, in whichn is a unit normal vector field alongf(V) andL,M,N are the corresponding components of the second fundamental form:

The key to this definition is that⁠∂f/∂u⁠,⁠∂f/∂v⁠, andn form a basis ofℝ³ at each point, relative to which each of the three equations uniquely specifies the Christoffel symbols as coordinates of the second partial derivatives off. The choice of unit normal has no effect on the Christoffel symbols, since ifn is exchanged for its negation, then the components of the second fundamental form are also negated, and so the signs ofLn,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 ofE,F,G.

TheCodazzi equations assert that^[23]

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 tov, the second equation with respect tou, subtracting the two, and taking thedot product withn. TheGauss equation asserts that^[24]

These can be similarly derived as the Codazzi equations, with one using theWeingarten equations instead of taking thedot product withn. 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 byBrioschi using a skillful use of determinants:^[25][26]

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 thatLN −M² can actually be written as a function ofE,F,G, even though the individual componentsL,M,N cannot. This is known as thetheorema egregium, and was a major discovery ofCarl Friedrich Gauss. It is particularly striking when one recalls the geometric definition of the Gaussian curvature ofS as being defined by the maximum and minimum radii of osculating circles; they seem to be fundamentally defined by the geometry of howS bends withinℝ³. Nevertheless, the theorem shows that their product can be determined from the "intrinsic" geometry ofS, having only to do with the lengths of curves alongS and the angles formed at their intersections. As said byMarcel 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 besuccinctly expressed and derived in the language ofconnection forms due toÉlie Cartan.^[28] In the language oftensor calculus, making use of natural metrics and connections ontensor bundles, the Gauss equation can be written asH² − |h|² =R and the two Codazzi equations can be written as∇₁h₁₂ = ∇₂h₁₁ and∇₁h₂₂ = ∇₂h₁₂; 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 thescalar curvatureR.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 generalRiemannian manifolds.

A diffeomorphism${\displaystyle \phi }$ between open sets${\displaystyle U}$ and${\displaystyle V}$ in a regular surface${\displaystyle S}$ is said to be anisometry 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

${\displaystyle E(p)w_1\cdot w_1+2F(p)w_1\cdot w_2+G(p)w_2\cdot w_2=E(\phi (p)){\phi }^{\mathrm{\prime }}(w_1)\cdot {\phi }^{\mathrm{\prime }}(w_1)+2F(\phi (p)){\phi }^{\mathrm{\prime }}(w_1)\cdot {\phi }^{\mathrm{\prime }}(w_2)+G(\phi (p)){\phi }^{\mathrm{\prime }}(w_1)\cdot {\phi }^{\mathrm{\prime }}(w_2).}$

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

${\displaystyle (w_1,w_2{)}_p=({\phi }^{\mathrm{\prime }}(w_1),{\phi }^{\mathrm{\prime }}(w_2){)}_{\phi (p)}}$.

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

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

Conversely if${\displaystyle \phi }$ preserves the lengths of all parametrized in curves then${\displaystyle \phi }$ 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${\displaystyle {\phi }^{\mathrm{\prime }}(w_1)}$ and${\displaystyle {\phi }^{\mathrm{\prime }}(w_2)}$, so that${\displaystyle ({\phi }^{\mathrm{\prime }}(w_1),{\phi }^{\mathrm{\prime }}(w_2){)}_{\phi (p)}=(w_1,w_1{)}_p}$.^[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${\displaystyle \phi =f_2\circ f_1^{-1}}$ 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, thecatenoid andhelicoid are locally isometric.^[36]

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Atangential vector fieldX onS assigns, to eachp inS, a tangent vectorX_p toS atp. According to the "intrinsic" definition of tangent vectors given above, a tangential vector fieldX then assigns, to each local parametrizationf :V →S, two real-valued functionsX¹ andX² onV, so that

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

for eachp inS. One says thatX is smooth if the functionsX¹ andX² are smooth, for any choice off.^[37] According to the other definitions of tangent vectors given above, one may also regard a tangential vector fieldX onS as a mapX :S → ℝ³ such thatX(p) is contained in the tangent spaceT_pS ⊂ ℝ³ for eachp inS. As is common in the more general situation ofsmooth manifolds, tangential vector fieldscan also be defined as certain differential operators on the space of smooth functions onS.

Thecovariant derivatives (also called "tangential derivatives") ofTullio Levi-Civita andGregorio Ricci-Curbastro provide a means of differentiating smooth tangential vector fields. Given a tangential vector fieldX and a tangent vectorY toS atp, the covariant derivative∇_YX is a certain tangent vector toS atp. Consequently, ifX andY are both tangential vector fields, then∇_YX can also be regarded as a tangential vector field; iteratively, ifX,Y, andZ 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 theChristoffel symbols and the "intrinsic" definition of tangent vectors, and the second is more manifestly geometric.

Given a tangential vector fieldX and a tangent vectorY toS atp, one defines∇_YX to be the tangent vector top which assigns to a local parametrizationf :V →S the two numbers

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

whereD_(Y¹,Y²) is thedirectional derivative.^[38] This is often abbreviated in the less cumbersome form(∇_YX)^k = ∂_Y(X^k) +YⁱΓ^k
_ijX ^j, making use ofEinstein notation and with the locations of function evaluation being implicitly understood. This follows astandard prescription inRiemannian geometry for obtaining aconnection from aRiemannian metric. It is a fundamental fact that the vector

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

inℝ³ is independent of the choice of local parametizationf, although this is rather tedious to check.

One can also define the covariant derivative by the following geometric approach, which does not make use ofChristoffel symbols or local parametrizations.^[39][40][41] LetX be a vector field onS, viewed as a functionS → ℝ³. Given any curvec : (a,b) →S, one may consider the compositionX ∘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) ofℝ³. Theorthogonal projection of this vector ontoT_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 onc′(t) andX, and not onc andX; 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 ofS; 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 regardsX¹⁠∂f/∂u⁠ +X²⁠∂f/∂v⁠ as aℝ³-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 off 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

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

can be recognized as the second coordinate of

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

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 ofRiemannian geometry, this observation can also be phrased as saying that the right-hand sides of the Gauss equations are various components of theRicci curvature of theLevi-Civita connection of the first fundamental form, when interpreted as aRiemannian metric.

A surface of revolution is obtained by rotating a curve in thexz-plane about thez-axis. Such surfaces include spheres, cylinders, cones, tori, and thecatenoid. The generalellipsoids,hyperboloids, andparaboloids are not. Suppose that the curve is parametrized by

${\displaystyle x=c_1(s),z=c_2(s)}$

withs drawn from an interval(a,b). Ifc₁ is never zero, ifc₁′ andc₂′ are never both equal to zero, and ifc₁ andc₂ are both smooth, then the corresponding surface of revolution

${\displaystyle S=\{(c_1(s)\cos t,c_1(s)\sin t,c_2(s)):s\in (a,b)\text{ and }t\in \mathbb{R}\}}$

will be a regular surface inℝ³. A local parametrizationf : (a,b) × (0, 2π) →S is given by

${\displaystyle f(s,t)=(c_1(s)\cos t,c_1(s)\sin t,c_2(s)).}$

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

Quantity	Formula
A unit normal vector field	${\displaystyle n=\frac{(-c_2^{\prime }(s)\cos t,-c_2^{\prime }(s)\sin t,c_1^{\prime }(s))}{\sqrt{c_1^{\prime }(s{)}^2+c_2^{\prime }(s{)}^2}}}$
First fundamental form
Second fundamental form
Principal curvatures	${\displaystyle \frac{c_1^{\prime }(s)c_2^{″}(s)-c_2^{\prime }(s)c_1^{″}(s)}{(c_1^{\prime }(s{)}^2+c_2^{\prime }(s{)}^2{)}^{3/2}}\text{and}\frac{c_2^{\prime }(s)}{c_1(s)\sqrt{c_1^{\prime }(s{)}^2+c_2^{\prime }(s{)}^2}}}$
Gaussian curvature	${\displaystyle K=\frac{c_2^{\prime }(s)(c_1^{\prime }(s)c_2^{″}(s)-c_2^{\prime }(s)c_1^{″}(s))}{c_1(s)(c_1^{\prime }(s{)}^2+c_2^{\prime }(s{)}^2{)}^2}}$
Mean curvature	${\displaystyle H=\frac{1}{2}\frac{c_1^{\prime }(s)c_2^{″}(s)-c_2^{\prime }(s)c_1^{″}(s)}{(c_1^{\prime }(s{)}^2+c_2^{\prime }(s{)}^2{)}^{3/2}}+\frac{1}{2}\frac{c_2^{\prime }(s)}{c_1(s)\sqrt{c_1^{\prime }(s{)}^2+c_2^{\prime }(s{)}^2}}.}$

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 findc₁′(s)c₁′′(s) +c₂′(s)c₂′′(s) = 0. On substitution into the Gaussian curvature, one has the simplified

${\displaystyle K=-\frac{c_1^{″}(s)}{c_1(s)}\text{and}H=c_1^{\prime }(s)c_2^{″}(s)-c_2^{\prime }(s)c_1^{″}(s)+\frac{c_2^{\prime }(s)}{c_1(s)}.}$

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 thatc₂(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 onS can be parametrized as a geodesic; a constant-s curve onS can be parametrized as a geodesic if and only ifc₁′(s) is equal to zero. Generally, geodesics onS are governed byClairaut's relation.

Consider the quadric surface defined by^[45]

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

This surface admits a parametrization

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

The Gaussian curvature and mean curvature are given by

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

A ruled surface is one which can be generated by the motion of a straight line inE³.^[46] Choosing adirectrix on the surface, i.e. a smooth unit speed curvec(t) orthogonal to the straight lines, and then choosingu(t) to be unit vectors along the curve in the direction of the lines, the velocity vectorv =c_t andu satisfy

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

The surface consists of points

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

ass andt vary.

Then, if

${\displaystyle a=\Vert u_t\Vert ,b=u_t\cdot v,\alpha =-\frac{b}{a^2},\beta =\frac{\sqrt{a^2-b^2}}{a^2},}$

the Gaussian and mean curvature are given by

${\displaystyle K=-\frac{{\beta }^2}{((s-\alpha {)}^2+{\beta }^2{)}^2},K_m=-\frac{r[(s-\alpha {)}^2+{\beta }^2)]+{\beta }_t(s-\alpha )+\beta {\alpha }_t}{[(s-\alpha {)}^2+{\beta }^2{]}^{\frac{3}{2}}}.}$

The Gaussian curvature of the ruled surface vanishes if and only ifu_t andv are proportional,^[47] This condition is equivalent to the surface being theenvelope of the planes along the curve containing the tangent vectorv and the orthogonal vectoru, i.e. to the surface beingdevelopable along the curve.^[48] More generally a surface inE³ 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.)

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

Given a closed curve inE³, find a surface having the curve as boundary with minimal area.

Such a surface is called aminimal surface.

In 1776Jean 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 calledPlateau's problem after the Belgian physicistJoseph Plateau who carried out experiments on soap films in the mid-nineteenth century. In 1930Jesse Douglas andTibor Radó gave an affirmative answer to Plateau's problem (Douglas was awarded one of the firstFields medals for this work in 1936).^[51]

Many explicit examples of minimal surface are known explicitly, such as thecatenoid, thehelicoid, theScherk surface and theEnneper surface. There has been extensive research in this area, summarised inOsserman (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 inS².

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

• The unitsphere inE³ has constant Gaussian curvature +1.
• The Euclideanplane and thecylinder both have constant Gaussian curvature 0.
• A unitpseudosphere has constant Gaussian curvature -1 (apart from its equator, that issingular). Pseudosphere can be obtained by rotating atractrix around its asymptote. In 1868Eugenio Beltrami showed that the geometry of the pseudosphere was directly related to that of the more abstracthyperbolic plane, discovered independently byLobachevsky (1830) andBolyai (1832). Already in 1840, F. Minding, a student of Gauss, had obtained trigonometric formulas for the pseudosphere identical to those for the hyperbolic plane.^[53] The intrinsic geometry of this surface is now better understood in terms of thePoincaré metric on theupper half plane or theunit disc, and has been described by other models such as theKlein model or thehyperboloid model, obtained by considering the two-sheeted hyperboloidq(x,y,z) = −1 in three-dimensionalMinkowski space, whereq(x,y,z) =x² +y² –z².^[54]

The sphere, the plane and the hyperbolic plane havetransitiveLie 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 toPoincaré'suniformization theorem (see below).

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

For any surface embedded in Euclidean space of dimension 3 or higher, it is possible to measure the length of a curve on the surface, the angle between two curves and the area of a region on the surface. This structure is encoded infinitesimally in aRiemannian metric on the surface throughline elements andarea elements. Classically in the nineteenth and early twentieth centuries only surfaces embedded inR³ were considered and the metric was given as a 2×2positive definite matrix varying smoothly from point to point in a local parametrization of the surface. The idea of local parametrization and change of coordinate was later formalized through the current abstract notion of amanifold, a topological space where thesmooth structure is given by local charts on the manifold, exactly as the planet Earth is mapped byatlases today. Changes of coordinates between different charts of the same region are required to be smooth. Just as contour lines on real-life maps encode changes in elevation, taking into account local distortions of the Earth's surface to calculate true distances, so the Riemannian metric describes distances and areas "in the small" in each local chart. In each local chart a Riemannian metric is given by smoothly assigning a 2×2 positive definite matrix to each point; when a different chart is taken, the matrix is transformed according to theJacobian matrix of the coordinate change. The manifold then has the structure of a 2-dimensionalRiemannian manifold.

Thedifferentialdn of theGauss mapn can be used to define a type of extrinsic curvature, known as theshape operator^[55] orWeingarten map. This operator first appeared implicitly in the work ofWilhelm Blaschke and later explicitly in a treatise by Burali-Forti and Burgati.^[56] Since at each pointx of the surface, the tangent space is aninner product space, the shape operatorS_x can be defined as a linear operator on this space by the formula

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

for tangent vectorsv,w (the inner product makes sense becausedn(v) andw both lie inE³).^[c] The right hand side is symmetric inv andw, so the shape operator isself-adjoint on the tangent space. The eigenvalues ofS_x are just the principal curvaturesk₁ andk₂ atx. In particular thedeterminant of the shape operator at a point is the Gaussian curvature, but it also contains other information, since themean curvature is half thetrace 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. Ifn is a unit normal field toM andv is a tangent vector then

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

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

In general, theeigenvectors and eigenvalues of the shape operator at each point determine the directions in which the surface bends at each point. The eigenvalues correspond to theprincipal 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.

Curves on a surface which minimize length between the endpoints are calledgeodesics; they are the shape that anelastic band stretched between the two points would take. Mathematically they are described usingordinary differential equations and thecalculus of variations. The differential geometry of surfaces revolves around the study of geodesics. It is still an open question whether every Riemannian metric on a 2-dimensional local chart arises from an embedding in 3-dimensional Euclidean space: the theory of geodesics has been used to show this is true in the important case when the components of the metric areanalytic.

Given a piecewise smooth pathc(t) = (x(t),y(t)) in the chart fort in[a,b], itslength is defined by

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

andenergy 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 theEuler–Lagrange equations, ifc(t) is a path minimising length,parametrized by arclength, it must satisfy theEuler equations

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

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

where theChristoffel symbolsΓ^k
_ij are given by

${\displaystyle {\mathrm{\Gamma }}_{ij}^k=\frac{1}{2}{g}^{km}({\mathrm{\partial }}_j{g}_{im}+{\mathrm{\partial }}_i{g}_{jm}-{\mathrm{\partial }}_m{g}_{ij})}$

whereg₁₁ =E,g₁₂ =F,g₂₂ =G andg^ij is the inverse matrix tog_ij. A path satisfying the Euler equations is called ageodesic. By theCauchy–Schwarz inequality a path minimising energy is just a geodesic parametrised by arc length; and, for any geodesic, the parametert is proportional to arclength.^[57]

Thegeodesic curvaturek_g at a point of a curvec(t), parametrised by arc length, on an oriented surface is defined to be^[58]

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

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