Second fundamental form

The second fundamental form is in mathematics a function from the differential geometry. Defines the second fundamental form was first in the theory of surfaces in three-dimensional space, a branch of classical differential geometry. Today, there are also a generalized definition in the Riemannian geometry.

During the first fundamental form, the inner geometry of a surface (that is, attributes that can be determined by distance measurements within the area ) describing the second fundamental form of the position of the surface depends on the surrounding space. It is required for curvature calculations and occurs, for example in the Mainardi - Codazzi equations. With their help and with the help of the first fundamental form of the principal curvatures, the mean curvature and the Gaussian curvature of the surface can be defined.

  • 2.1 Definition
  • 2.2 Features
  • 2.3 Scalar second fundamental form

Classical differential geometry

Definition

A surface is given by, ie by and parameterized. Is the first fundamental form of the surface positive-definite, so you can assign a unit normal vector of the surface. For the parameter values ​​and by the particular point in the area of ​​this is due to the vector product

Given. The coefficients of the second fundamental form in this point are defined as follows:

Defined. Here, and the second partial derivatives with respect to the parameters. The Malpunkte express inner products of vectors. To simplify the notation we often leaves the arguments away and writes only, and. Some authors use the terms, and.

The second fundamental form is then the quadratic form

Occasionally, the notation is used with differentials:

Another ( more modern ) spelling is:

So the second fundamental form has the matrix representation

Frequently, called the second fundamental form and the bilinear form represented by this matrix.

Properties

The discriminant (the determinant of the matrix representation ) of the second fundamental form provides information on how the given surface is curved at the point considered. Three cases can be distinguished:

  • For there is elliptical curvature. (Example: surface of an ellipsoid or sphere )
  • Means parabolic curvature. ( For example, the surface of a right circular cylinder )
  • If true, this is called hyperbolic curvature. (Example: Single-leaf hyperboloid )

Example

Following the example from the article of the first fundamental form following, the surface of a sphere of radius is considered again. This area is again

Parameterized. The unit normal field can then

Are described. The second partial derivatives of noisy

Therefore, we obtain the coefficients and. The representation of the second fundamental form of the sphere with the help of differentials is then

Riemannian geometry

In contrast to the first fundamental form, which was in the Riemannian geometry replaced by more descriptive constructions, the second fundamental form in the Riemannian geometry has an important meaning and a generalized definition.

Definition

Be a submanifold of the Riemannian manifold starting point for the definition of the second fundamental form is the orthogonal decomposition of vector fields in into tangential and normal components. Are vector fields, so you can continue to these vector fields on. If the Levi- Civita connection on, then we obtain the decomposition

The second fundamental form is a mapping

Which by

Is defined. This refers to the normal set of which is defined analogously to the tangent and the orthogonal projection onto the normal bundle.

Properties

The second fundamental form is

  • Regardless of the continuation of the vector fields and.
  • Bilinear over
  • Symmetrical and

Scalar second fundamental form

Let be a -dimensional Riemannian manifold with Riemannian metric and is a - dimensional submanifold of. Thus a submanifold of codimension 1 is also called hypersurface. In this case, the normal space is one-dimensional at each point of and there are exactly two unit normal vectors that span each. These differ only in sign.

Is a unit normal vector field fixed is selected, we define the associated scalar second fundamental form by

The scalar second fundamental form does not depend up on the sign of the choice of the unit normal vector field from: Taking place the second oppositely oriented unit normal vector field, so changes in the scalar second fundamental form, only the sign. From the properties of the second fundamental form follows that the scalar second fundamental form is also symmetric and linear in each argument, ie a symmetric (0,2) tensor field - on.

Total geodesic submanifolds

A submanifold is totally geodesic (ie geodesics in geodesics in are even ) if and only if its second fundamental form vanishes identically.

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