Second fundamental form#Generalization to arbitrary codimension

File:Second fundamental form.svg

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

: $z=L\backslash frac\{x^2\}\{2\}\; +\; Mxy\; +\; N\backslash frac\{y^2\}\{2\}\; +\; \backslash text\{higher\; order\; terms\}\backslash ,,$

and the second fundamental form at the origin in the coordinates {{math|(x,y)}} is the quadratic form

: $L\; \backslash ,\; dx^2\; +\; 2M\; \backslash ,\; dx\; \backslash ,\; dy\; +\; N\; \backslash ,\; dy^2\; \backslash ,.$

For a smooth point {{math|P}} on {{math|S}}, one can choose the coordinate system so that the plane {{math|z {{=}} 0}} is tangent to {{math|S}} at {{math|P}}, and define the second fundamental form in the same way.

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

:$\backslash mathbf\{n\}\; =\; \backslash frac\{\backslash mathbf\{r\}\_u\backslash times\backslash mathbf\{r\}\_v\}$

The second fundamental form is usually written as

:$\backslash mathrm\{I\backslash !I\}\; =\; L\backslash ,\; du^2\; +\; 2M\backslash ,\; du\backslash ,\; dv\; +\; N\backslash ,\; dv^2\; \backslash ,,$

its matrix in the basis {{math|{r_u, r_v}}} of the tangent plane is

:$\backslash begin\{bmatrix\}$

L&M\\

M&N

\end{bmatrix} \,.

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

:$L\; =\; \backslash mathbf\{r\}\_\{uu\}\; \backslash cdot\; \backslash mathbf\{n\}\backslash ,,\; \backslash quad$

M = \mathbf{r}_{uv} \cdot \mathbf{n}\,, \quad

N = \mathbf{r}_{vv} \cdot \mathbf{n}\,.

For a signed distance field of Hessian {{math|H}}, the second fundamental form coefficients can be computed as follows:

:$L\; =\; -\backslash mathbf\{r\}\_u\; \backslash cdot\; \backslash mathbf\{H\}\; \backslash cdot\; \backslash mathbf\{r\}\_u\backslash ,,\; \backslash quad$

M = -\mathbf{r}_u \cdot \mathbf{H} \cdot \mathbf{r}_v\,, \quad

N = -\mathbf{r}_v \cdot \mathbf{H} \cdot \mathbf{r}_v\,.

The second fundamental form of a general parametric surface {{math|S}} is defined as follows.

Let {{math|r {{=}} r(u¹,u²)}} be a regular parametrization of a surface in {{math|R³}}, where {{math|r}} is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of {{math|r}} with respect to {{math|u^α}} by {{math|r_α}}, {{math|α {{=}} 1, 2}}. Regularity of the parametrization means that {{math|r₁}} and {{math|r₂}} are linearly independent for any {{math|(u¹,u²)}} in the domain of {{math|r}}, and hence span the tangent plane to {{math|S}} at each point. Equivalently, the cross product {{math|r₁ × r₂}} is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors {{math|n}}:

:$\backslash mathbf\{n\}\; =\; \backslash frac\{\backslash mathbf\{r\}\_1\backslash times\backslash mathbf\{r\}\_2\}$

The second fundamental form is usually written as

:$\backslash mathrm\{I\backslash !I\}\; =\; b\_\{\backslash alpha\; \backslash beta\}\; \backslash ,\; du^\{\backslash alpha\}\; \backslash ,\; du^\{\backslash beta\}\; \backslash ,.$

The equation above uses the Einstein summation convention.

The coefficients {{math|b_αβ}} at a given point in the parametric {{math|u¹u²}}-plane are given by the projections of the second partial derivatives of {{math|r}} at that point onto the normal line to {{math|S}} and can be computed in terms of the normal vector {{math|n}} as follows:

:$b\_\{\backslash alpha\; \backslash beta\}\; =\; r\_\{,\backslash alpha\; \backslash beta\}^\{\backslash \; \backslash \; \backslash ,\backslash gamma\}\; n\_\{\backslash gamma\}\backslash ,.$

In Euclidean space, the second fundamental form is given by

:$\backslash mathrm\{I\backslash !I\}(v,w)\; =\; -\backslash langle\; d\backslash nu(v),w\backslash rangle\backslash nu$

where $\backslash nu$ is the Gauss map, and $d\backslash nu$ the differential of $\backslash nu$ 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 {{math|S}}) of a hypersurface,

:$\backslash mathrm\; I\backslash !\backslash mathrm\; I(v,w)=\backslash langle\; S(v),w\backslash rangle\; =\; -\backslash langle\; \backslash nabla\_v\; n,w\backslash rangle=\backslash langle\; n,\backslash nabla\_v\; w\backslash rangle\; \backslash ,,$

where {{math|∇_vw}} denotes the covariant derivative of the ambient manifold and {{math|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 {{math|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).

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

:$\backslash mathrm\{I\backslash !I\}(v,w)=(\backslash nabla\_v\; w)^\backslash bot\backslash ,,$

where $(\backslash nabla\_v\; w)^\backslash bot$ denotes the orthogonal projection of covariant derivative $\backslash nabla\_v\; w$ onto the normal bundle.

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

:$\backslash langle\; R(u,v)w,z\backslash rangle\; =\backslash mathrm\; I\backslash !\backslash mathrm\; I(u,z)\backslash mathrm\; I\backslash !\backslash mathrm\; I(v,w)-\backslash mathrm\; I\backslash !\backslash mathrm\; I(u,w)\backslash mathrm\; I\backslash !\backslash mathrm\; I(v,z).$

This is called the Gauss equation, as it may be viewed as a generalization of Gauss's Theorema Egregium.

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

:$\backslash langle\; R\_N(u,v)w,z\backslash rangle\; =\; \backslash langle\; R\_M(u,v)w,z\backslash rangle+\backslash langle\; \backslash mathrm\; I\backslash !\backslash mathrm\; I(u,z),\backslash mathrm\; I\backslash !\backslash mathrm\; I(v,w)\backslash rangle-\backslash langle\; \backslash mathrm\; I\backslash !\backslash mathrm\; I(u,w),\backslash mathrm\; I\backslash !\backslash mathrm\; I(v,z)\backslash rangle\backslash ,.$

• First fundamental form
• Gaussian curvature
• Gauss–Codazzi equations
• Shape operator
• Third fundamental form
• Tautological one-form

• {{cite book|first=Heinrich|last=Guggenheimer|title=Differential Geometry|year=1977|publisher=Dover|chapter=Chapter 10. Surfaces|isbn=0-486-63433-7}}
• {{cite book |author1=Kobayashi, Shoshichi |author2=Nomizu, Katsumi |name-list-style=amp | title = Foundations of Differential Geometry, Vol. 2 | publisher=Wiley-Interscience | year=1996 |edition=New |isbn = 0-471-15732-5}}
• {{cite book|last=Spivak|first=Michael|title=A Comprehensive introduction to differential geometry (Volume 3)|year=1999|publisher=Publish or Perish|isbn=0-914098-72-1}}

• Steven Verpoort (2008) [https://lirias.kuleuven.be/bitstream/1979/1779/2/hierrrissiedan!.pdf Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects] from Katholieke Universiteit Leuven.

{{curvature}}

Category:Differential geometry

Category:Differential geometry of surfaces

Category:Riemannian geometry

Category:Curvature (mathematics)

Category:Tensors