- Second fundamental form
In

differential geometry , the**second fundamental form**is aquadratic form on thetangent plane of a smooth surface in the three dimensionalEuclidean space , usually denoted by II. Together with thefirst fundamental form , it serves to define extrinsic invariants of the surface, itsprincipal curvature s. More generally, such a quadratic form is defined for a smoothhypersurface in aRiemannian manifold and a smooth choice of the unit normal vector at each point.**Surface in****R**^{3}**Motivation**The second fundamental form of a

parametric surface "S" in**R**^{3}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 istangent to the surface at the origin. Then "f" and itspartial derivative s with respect to "x" and "y" vanish at (0,0). Therefore, theTaylor 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**R**^{3}, where**r**is a smoothvector valued function of two variables. It is common to denote the partial derivatives of**r**with respect to "u" and "v" by**r**_{u}and**r**_{v}. Regularity of the parametrization means that**r**_{u}and**r**_{v}are linearly independent for any ("u","v") in the domain of**r**, and hence span the tangent plane to "S" at each point. Equivalently, thecross product **r**_{u}×**r**_{v}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 {

**r**_{u},**r**_{v}} of the tangent plane is:$egin\{bmatrix\}LM\backslash MNend\{bmatrix\}.$

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 thedot 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**R**^{3}, where**r**is a smoothvector valued function of two variables. It is common to denote the partial derivatives of**r**with respect to "u"^{α}by**r**_{α}, α = 1, 2. Regularity of the parametrization means that**r**_{1}and**r**_{1}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 thedot 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 avector valued differential form , and the brackets denote themetric 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 theaffine 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 thenormal 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 , thecurvature tensor of asubmanifold 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'sTheorema Egregium . Theeigenvalue s of the second fundamental form, represented in anorthonormal basis , are theof the surface. A collection of orthonormalprincipal curvature seigenvector s 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.*