second fundamental form

In classical differential geometry the second fundamental formMathworldPlanetmath is a symmetric bilinear formMathworldPlanetmath defined on a differentiableMathworldPlanetmathPlanetmath surface M embedded in 3, which in some sense measures the curvatureMathworldPlanetmathPlanetmath of M in space.

To construct the second fundamental form requires a small digression. After the digression we will discuss how it relates to the curvature of M.

Construction of the second fundamental form

Consider the tangent planesMathworldPlanetmath TpM of the surface M for each point pM. There are two unit normalsMathworldPlanetmath to TpM. Assuming M is orientable, we can choose one of these unit normals (, n(p), so that n(p) varies smoothly with p.

Since n(p) is a unit vectorMathworldPlanetmath in 3, it may be considered as a point on the sphere S23. Then we have a map n:MS2. It is called the normal map or Gauss map.

The second fundamental form is the tensor field on M defined by

p(ξ,η)=-Dnp(ξ),η,ξ,ηTpM, (1)

where , is the dot productMathworldPlanetmath of 3, and we consider the tangent planes of surfaces in 3 to be subspacesPlanetmathPlanetmathPlanetmath of 3.

The linear transformation Dnp is in reality the tangentMathworldPlanetmathPlanetmathPlanetmath mapping Dnp:TpMTn(p)S2, but since Tn(p)S2=TpM by the definition of n, we prefer to think of Dnp as Dnp:TpMTpM.

The tangent map Dn, is often called the Weingarten map.

Proposition 1.

The second fundamental form is a symmetric form.


This is a computation using a coordinate chart σ for M. Let u,v be the corresponding names for the coordinatesMathworldPlanetmathPlanetmath. From the equation


differentiating with respect to u using the product ruleMathworldPlanetmath gives

n,2σuv=-nu,σv=-Dn(σu),σv=(σu,σv). (2)

(The second equality follows from the definition of the tangent map Dn.) Reversing the roles of u,v and repeating the last derivation, we obtain also:

n,2σuv=n,2σvu=(σv,σu). (3)

Since σ/u and σ/v form a basis for TpM, combining (2) and (3) proves that is symmetricMathworldPlanetmathPlanetmathPlanetmathPlanetmath. ∎

In view of PropositionPlanetmathPlanetmath 1, it is customary to regard the second fundamental form as a quadratic formMathworldPlanetmath, as it done with the first fundamental formMathworldPlanetmath. Thus, the second fundamental form is referred to with the following expression11 Unfortunately the coefficient M here clashes with our use of the letter M for the surface (manifold), but whenever we write M, the context should make clear which meaning is intended. The use of the symbols L,M,N for the coefficients of the second fundamental form is standard, but probably was established long before anyone thought about manifolds.:


Compare with the tensor notation


Or in matrix form (with respect to the coordinates u,v),


Curvature of curves on a surface

Let γ be a curve lying on the surface M, parameterized by arc-length. Recall that the curvature κ(s) of γ at s is γ′′(s). If we want to measure the curvature of the surface, it is natural to consider the component of γ′′(s) in the normal n(γ(s)). Precisely, this quantity is


and is called the normal curvatureMathworldPlanetmathPlanetmathPlanetmath of γ on M.

So to study the curvature of M, we ignore the component of the curvature of γ in the tangent plane of M. Also, physically speaking, the normal curvature is proportional to the acceleration required to keep a moving particle on the surface M.

We now come to the motivation for defining the second fundamental form:

Proposition 2.

Let γ be a curve on M, parameterized by arc-length, and γ(s)=p. Then


From the equation


differentiate with respect to s:

n(γ(s)),γ′′(s) =-ddsn(γ(s)),γ(s)

It is now time to mention an important consequence of Proposition 1: the fact that is symmetric means that -Dn is self-adjointPlanetmathPlanetmath with respect to the inner product (the first fundamental form). So, if -Dn is expressed as a matrix with orthonormal coordinates (with respect to ), then the matrix is symmetric. (The minus sign in front of Dn is to make the formulasMathworldPlanetmathPlanetmath work out nicely.)

Certain theorems in linear algebra tell us that, -Dnp being self-adjoint, it has an orthonormal basis of eigenvectorsMathworldPlanetmathPlanetmathPlanetmath e1,e2 with corresponding eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath κ1κ2. These eigenvalues are called the principal curvatures of M at p. The eigenvectors e1,e2 are the principal directions. The terminology is justified by the following theorem:

Theorem 1 (Euler’s Theorem).

The normal curvature of a curve γ has the form


It follows that the minimum possible normal curvature is κ1, and the maximum possible is κ2.


Since e1,e2 form an orthonormal basis for TpM, we may write


for some angle θ. Then

γ′′(s),n(p) =(γ(s),γ(s))

Matrix representations of second fundamental form and Weingarten map

At this point, we should find the explicit prescriptions for calculating the second fundamental form and the Weingarten map.

Let σ be a coordinate chart for M, and u,v be the names of the coordinates. For a test vector ξTpM, we write ξu and ξv for the u,v coordinates of ξ.

We compute the matrix W for -Dn in u,v-coordinates. We have

(ξuξv)(LMMN)(ξuξv) =(ξ,ξ)=-Dn(ξ),ξ

where Q is the matrix that changes from u,v-coordinates to orthonormal coordinates for TpM — this is necessary to compute the inner product. But

QTQ=(EFFG)=(the first fundamental form),

because Q is the matrix with columns σ/u and σ/v expressed in orthonormal coordinates.

(More to be written…)


Title second fundamental form
Canonical name SecondFundamentalForm
Date of creation 2013-03-22 15:29:02
Last modified on 2013-03-22 15:29:02
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 5
Author stevecheng (10074)
Entry type Definition
Classification msc 53A05
Related topic FirstFundamentalForm
Related topic ShapeOperator
Related topic NormalSection
Related topic NormalCurvatures
Defines normal curvature
Defines principal direction
Defines principal curvature
Defines Weingarten matrix
Defines Weingarten map
Defines Gauss map