second fundamental form
To construct the second fundamental form requires a small digression. After the digression we will discuss how it relates to the curvature of .
Construction of the second fundamental form
Consider the tangent planes of the surface for each point . There are two unit normals to . Assuming is orientable, we can choose one of these unit normals (http://planetmath.org/MutualPositionsOfVectors), , so that varies smoothly with .
Since is a unit vector in , it may be considered as a point on the sphere . Then we have a map . It is called the normal map or Gauss map.
The second fundamental form is the tensor field on defined by
The tangent map , is often called the Weingarten map.
The second fundamental form is a symmetric form.
differentiating with respect to using the product rule gives
(The second equality follows from the definition of the tangent map .) Reversing the roles of and repeating the last derivation, we obtain also:
In view of Proposition 1, it is customary to regard the second fundamental form as a quadratic form, as it done with the first fundamental form. Thus, the second fundamental form is referred to with the following expression11 Unfortunately the coefficient here clashes with our use of the letter for the surface (manifold), but whenever we write , the context should make clear which meaning is intended. The use of the symbols for the coefficients of the second fundamental form is standard, but probably was established long before anyone thought about manifolds.:
Compare with the tensor notation
Or in matrix form (with respect to the coordinates ),
Curvature of curves on a surface
Let be a curve lying on the surface , parameterized by arc-length. Recall that the curvature of at is . If we want to measure the curvature of the surface, it is natural to consider the component of in the normal . Precisely, this quantity is
and is called the normal curvature of on .
So to study the curvature of , we ignore the component of the curvature of in the tangent plane of . Also, physically speaking, the normal curvature is proportional to the acceleration required to keep a moving particle on the surface .
We now come to the motivation for defining the second fundamental form:
Let be a curve on , parameterized by arc-length, and . Then
From the equation
differentiate with respect to :
It is now time to mention an important consequence of Proposition 1: the fact that is symmetric means that is self-adjoint with respect to the inner product (the first fundamental form). So, if is expressed as a matrix with orthonormal coordinates (with respect to ), then the matrix is symmetric. (The minus sign in front of is to make the formulas work out nicely.)
Certain theorems in linear algebra tell us that, being self-adjoint, it has an orthonormal basis of eigenvectors with corresponding eigenvalues . These eigenvalues are called the principal curvatures of at . The eigenvectors are the principal directions. The terminology is justified by the following theorem:
Matrix representations of second fundamental form and Weingarten map
At this point, we should find the explicit prescriptions for calculating the second fundamental form and the Weingarten map.
Let be a coordinate chart for , and be the names of the coordinates. For a test vector , we write and for the coordinates of .
We compute the matrix for in -coordinates. We have
where is the matrix that changes from -coordinates to orthonormal coordinates for — this is necessary to compute the inner product. But
because is the matrix with columns and expressed in orthonormal coordinates.
(More to be written…)
|Title||second fundamental form|
|Date of creation||2013-03-22 15:29:02|
|Last modified on||2013-03-22 15:29:02|
|Last modified by||stevecheng (10074)|