# Second fundamental form

In differential geometry, the **second fundamental form** (or **shape tensor**) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by (read "two"). 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.

## Contents

## Surface in R^{3}[edit]

### Motivation[edit]

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 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:

and the second fundamental form at the origin in the coordinates (*x*,*y*) is the quadratic form

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[edit]

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 smooth vector-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, the cross product **r**_{u} × **r**_{v} is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors **n**:

The second fundamental form is usually written as

its matrix in the basis {**r**_{u}, **r**_{v}} of the tangent plane is

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:

### Physicist's notation[edit]

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 smooth vector-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**_{2} 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 **r**_{1} × **r**_{2} is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors **n**:

The second fundamental form is usually written as

The equation above uses the Einstein summation convention.

The coefficients *b*_{αβ} 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 in terms of the normal vector **n** as follows:

## Hypersurface in a Riemannian manifold[edit]

In Euclidean space, the second fundamental form is given by

where *ν* is the Gauss map, and *dν* the differential of *ν* 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,

where ∇_{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[edit]

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

where (∇_{v}*w*)^{⊥} denotes the orthogonal projection of covariant derivative ∇_{v}*w* onto the normal bundle.

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

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 *N* is a manifold embedded in a Riemannian manifold (*M*,*g*) then the curvature tensor *R _{N}* of

*N*with induced metric can be expressed using the second fundamental form and

*R*, the curvature tensor of

_{M}*M*:

## See also[edit]

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

## References[edit]

- Guggenheimer, Heinrich (1977). "Chapter 10. Surfaces".
*Differential Geometry*. Dover. ISBN 0-486-63433-7. - Kobayashi, Shoshichi & Nomizu, Katsumi (1996).
*Foundations of Differential Geometry, Vol. 2*(New ed.). Wiley-Interscience. ISBN 0-471-15732-5. - Spivak, Michael (1999).
*A Comprehensive introduction to differential geometry (Volume 3)*. Publish or Perish. ISBN 0-914098-72-1.

## External links[edit]

- Steven Verpoort (2008) Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects from Katholieke Universiteit Leuven.