Second fundamental form

Second fundamental form

In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a smooth surface in the three dimensional Euclidean space, usually denoted by II. Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth hypersurface in a Riemannian manifold and a smooth choice of the unit normal vector at each point.

Surface in R3

Motivation

The second fundamental form of a parametric surface "S" in R3 was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, "z" = "f"("x","y"), and that the plane "z" = 0 is tangent to the surface at the origin. Then "f" and its partial derivatives with respect to "x" and "y" vanish at (0,0). Therefore, the Taylor expansion of "f" at (0,0) starts with quadratic terms:

: z=Lfrac{x^2}{2} + Mxy + Nfrac{y^2}{2} + mathrm{scriptstyle }higher{ }order{ }terms,

and the second fundamental form at the origin in the coordinates "x", "y" is the quadratic form

: L dx^2 + 2M dx dy + N dy^2. ,

For a smooth point "P" on "S", one can choose the coordinate system so that the coordinate "z"-plane is tangent to "S" at "P" and define the second fundamental form in the same way.

Classical notation

The second fundamental form of a general parametric surface is defined as follows. Let r=r("u","v") be a regular parametrization of a surface in R3, where r is a smooth vector valued function of two variables. It is common to denote the partial derivatives of r with respect to "u" and "v" by ru and rv. Regularity of the parametrization means that ru and rv are linearly independent for any ("u","v") in the domain of r, and hence span the tangent plane to "S" at each point. Equivalently, the cross product ru × rv is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:

:mathbf{n} = frac{mathbf{r}_u imesmathbf{r}_v}.

The second fundamental form is usually written as

:mathrm{II} = Ldu^2 + 2Mdudv + Ndv^2, ,

its matrix in the basis {ru, rv} of the tangent plane is

: egin{bmatrix}L&M\M&Nend{bmatrix}.

The coefficients "L", "M", "N" at a given point in the parametric "uv"-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to "S" and can be computed with the aid of the dot product as follows:

:L = mathbf{r}_{uu} cdot mathbf{n}, quadM = mathbf{r}_{uv} cdot mathbf{n}, quadN = mathbf{r}_{vv} cdot mathbf{n}.

Modern notation

The second fundamental form of a general parametric surface "S" is defined as follows: Let r=r("u"1,"u"2) be a regular parametrization of a surface in R3, where r is a smooth vector valued function of two variables. It is common to denote the partial derivatives of r with respect to "u"α by rα, α = 1, 2. Regularity of the parametrization means that r1 and r1 are linearly independent for any ("u"1,"u"2) in the domain of r, and hence span the tangent plane to "S" at each point. Equivalently, the cross product r1 × r2 is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:

:mathbf{n} = frac{mathbf{r}_1 imesmathbf{r}_2}.

The second fundamental form is usually written as

:mathrm{II} = b_{alpha, eta} du^{alpha} du^{eta}. ,

The equation above implies Einstein Summation Convention. The coefficients "b"α,β at a given point in the parametric ("u"1, "u"2)-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to "S" and can be computed with the aid of the dot product as follows:

:b_{alpha, eta} = mathbf{r}_{alpha, eta} cdot mathbf{n}.

Hypersurface in a Riemannian manifold

In Euclidean space, the second fundamental form is given by

:I!I(v,w) = langle d u(v),w angle

where u is the Gauss map, and d u the differential of u regarded as a vector valued differential form, and the brackets denote the metric tensor of Euclidean space.

More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the shape operator (denoted by "S") of a hypersurface,

:mathrm I!mathrm I(v,w)=langle S(v),w angle= -langle abla_v n,w angle=langle n, abla_v w angle,

where abla_v w denotes the covariant derivative of the ambient manifold and "n" a field of normal vectors on the hypersurface. (If the affine connection is torsion-free, then the second fundamental form is symmetric.)

The sign of the second fundamental form depends on the choice of direction of "n" (which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of orientation of the surface).

Generalization to arbitrary codimension

The second fundamental form can be generalized to arbitrary codimension. In that case it is a quadratic form on the tangent space with values in the normal bundle and it can be defined by

:mathrm{I}!mathrm{I}(v,w)=( abla_v w)^ot,

where ( abla_v w)^ot denotes the orthogonal projection of covariant derivative abla_v w onto the normal bundle.

In Euclidean space, the curvature tensor of a submanifold can be described by the following formula:

:langle R(u,v)w,z angle =langle mathrm I!mathrm I(u,z),mathrm I!mathrm I(v,w) angle-langle mathrm I!mathrm I(u,w),mathrm I!mathrm I(v,z) angle.

This is called the Gauss equation, as it may be viewed as a generalization of Gauss's Theorema Egregium. The eigenvalues of the second fundamental form, represented in an orthonormal basis, are the principal curvatures of the surface. A collection of orthonormal eigenvectors are called the principal directions.

For general Riemannian manifolds one has to add the curvature of ambient space; if "N" is a manifold embedded in a Riemannian manifold ("M,g") then the curvature tensor R_N of "N" with induced metric can be expressedusing the second fundamental form and R_M , the curvature tensor of "M":

:langle R_N(u,v)w,z angle = langle R_M(u,v)w,z angle+langle mathrm I!mathrm I(u,z),mathrm I!mathrm I(v,w) angle-langle mathrm I!mathrm I(u,w),mathrm I!mathrm I(v,z) angle.

ee also

*First fundamental form
*Gaussian curvature
*Gauss–Codazzi equations

References

*
*cite book | author=Kobayashi, Shoshichi and Nomizu, Katsumi | title = Foundations of Differential Geometry, Vol. 2 | publisher=Wiley-Interscience | year=1996 (New edition) |id = ISBN 0471157325
*

External links

* A PHD thesis about the geometry of the second fundamental form by Steven Verpoort: https://repository.libis.kuleuven.be/dspace/bitstream/1979/1779/2/hierrrissiedan!.pdf


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • First fundamental form — In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three dimensional Euclidean space which is induced canonically from the dot product of R 3 . It permits the calculation of curvature… …   Wikipedia

  • Fundamental theorem of calculus — The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration.The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that… …   Wikipedia

  • Second law of thermodynamics — The second law of thermodynamics is an expression of the universal law of increasing entropy, stating that the entropy of an isolated system which is not in equilibrium will tend to increase over time, approaching a maximum value at… …   Wikipedia

  • Second language acquisition — is the process by which people learn a second language in addition to their native language(s). The term second language is used to describe the acquisition of any language after the acquisition of the mother tongue. The language to be learned is …   Wikipedia

  • Second Coming of Christ — Second Coming redirects here. For other uses, see Second Coming (disambiguation). Part of a series on Eschatology …   Wikipedia

  • Form — • The original meaning of the term form, both in Greek and Latin, was and is that in common use • eidos, being translated, that which is seen, shape, etc., with secondary meanings derived from this, as form, sort, particular, kind, nature… …   Catholic encyclopedia

  • Second harmonic generation — (SHG; also called frequency doubling) is a nonlinear optical process, in which photons interacting with a nonlinear material are effectively combined to form new photons with twice the energy, and therefore twice the frequency and half the… …   Wikipedia

  • Fundamental theorem of algebra — In mathematics, the fundamental theorem of algebra states that every non constant single variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed.Sometimes,… …   Wikipedia

  • Fundamental polygon — In mathematics, each closed surface in the sense of geometric topology can be constructed from an even sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges. Fundamental parallelogram defined by a pair of… …   Wikipedia

  • Second — This article is about the unit of time. For other uses, see Second (disambiguation). A light flashing approximately once per second. The second (SI unit symbol: s; informal abbreviation: sec) is a unit of measurement of time, and is the… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”