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").

Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form).

The second fundamental form of a parametric surface

The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures.

From a modern perspective, this theorem follows from the spectral theorem because these directions are as the principal axes of a symmetric tensor—the second fundamental form.

Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures.

where L, M, and N are the coefficients of the second fundamental form.

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

The first equation, sometimes called the Gauss equation (named after Carl Friedrich Gauss), relates the intrinsic curvature (or Gauss curvature) of the surface to the derivatives of the Gauss map, via the second fundamental form.

Unlike the second fundamental form, it is independent of the surface normal.

Equivalently, the determinant of the second fundamental form of a surface in

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

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").

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").

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").

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.

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.

was introduced and studied by Gauss.

First suppose that the surface is the graph of a twice continuously differentiable function,

is tangent to the surface at the origin.

and its partial derivatives with respect to

Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms:

is a smooth vector-valued function of two variables.

Equivalently, the cross product

and can be computed with the aid of the dot product as follows:

The equation above uses the Einstein summation convention.

is the Gauss map, and

the differential of

regarded as a vector-valued differential form, and the brackets denote the metric tensor of Euclidean space.