Second fundamental form
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In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, the second fundamental form (or shape tensor) is a
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
on the
tangent plane In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
of a
smooth surface In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensiv ...
in the three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, usually denoted by \mathrm (read "two"). Together with the
first fundamental form In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of . It permits the calculation of curvature and ...
, it serves to define extrinsic invariants of the surface, its
principal curvature In differential geometry, the two principal curvatures at a given point of a surface (mathematics), surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how ...
s. More generally, such a quadratic form is defined for a smooth immersed
submanifold In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...
in a
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
.


Surface in R3


Motivation

The second fundamental form of a
parametric surface A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that oc ...
in was introduced and studied by
Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
. First suppose that the surface is the graph of a twice
continuously differentiable In mathematics, a differentiable function of one Real number, real variable is a Function (mathematics), function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differ ...
function, , and that the plane is
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to the surface at the origin. Then and its
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...
s with respect to and vanish at (0,0). Therefore, the
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of ''f'' at (0,0) starts with quadratic terms: : z=L\frac + Mxy + N\frac + \text\,, and the second fundamental form at the origin in the coordinates is the
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
: L \, dx^2 + 2M \, dx \, dy + N \, dy^2 \,. For a smooth point on , one can choose the coordinate system so that the plane is tangent to at , and define the second fundamental form in the same way.


Classical notation

The second fundamental form of a general parametric surface is defined as follows. Let be a regular parametrization of a surface in , where is a smooth
vector-valued function A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
of two variables. It is common to denote the partial derivatives of with respect to and by and . Regularity of the parametrization means that and are linearly independent for any in the domain of , and hence span the tangent plane to at each point. Equivalently, the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors : :\mathbf = \frac \,. The second fundamental form is usually written as :\mathrm = L\, du^2 + 2M\, du\, dv + N\, dv^2 \,, its matrix in the basis of the tangent plane is : \begin L&M\\ M&N \end \,. The coefficients at a given point in the parametric -plane are given by the projections of the second partial derivatives of at that point onto the normal line to and can be computed with the aid of the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
as follows: :L = \mathbf_ \cdot \mathbf\,, \quad M = \mathbf_ \cdot \mathbf\,, \quad N = \mathbf_ \cdot \mathbf\,. For a signed distance field of Hessian , the second fundamental form coefficients can be computed as follows: :L = -\mathbf_u \cdot \mathbf \cdot \mathbf_u\,, \quad M = -\mathbf_u \cdot \mathbf \cdot \mathbf_v\,, \quad N = -\mathbf_v \cdot \mathbf \cdot \mathbf_v\,.


Physicist's notation

The second fundamental form of a general parametric surface is defined as follows. Let be a regular parametrization of a surface in , where is a smooth
vector-valued function A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
of two variables. It is common to denote the partial derivatives of with respect to by , . Regularity of the parametrization means that and are linearly independent for any in the domain of , and hence span the tangent plane to at each point. Equivalently, the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors : :\mathbf = \frac\,. The second fundamental form is usually written as :\mathrm = b_ \, du^ \, du^ \,. The equation above uses the
Einstein summation convention In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies s ...
. The coefficients at a given point in the parametric -plane are given by the projections of the second partial derivatives of at that point onto the normal line to and can be computed in terms of the normal vector as follows: :b_ = r_^ n_\,.


Hypersurface in a Riemannian manifold

In
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, the second fundamental form is given by :\mathrm(v,w) = -\langle d\nu(v),w\rangle\nu where \nu is the
Gauss map In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to its normal direction, a unit vector that is orthogonal to the surface at that point. Namely, given a surface ''X'' in Euclidean space R3 ...
, and d\nu the differential of \nu regarded as a vector-valued differential form, and the brackets denote the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
of Euclidean space. More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the
shape operator In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...
(denoted by ) of a hypersurface, :\mathrm I\!\mathrm I(v,w)=\langle S(v),w\rangle = -\langle \nabla_v n,w\rangle=\langle n,\nabla_v w\rangle \,, where denotes the
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
of the ambient manifold and a field of normal vectors on the hypersurface. (If the
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
is torsion-free, then the second fundamental form is symmetric.) The sign of the second fundamental form depends on the choice of direction of (which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building des ...
of the surface).


Generalization to arbitrary codimension

The second fundamental form can be generalized to arbitrary
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals ...
. In that case it is a quadratic form on the tangent space with values in the
normal bundle In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Definition Riemannian manifold Let (M,g) be a Riemannian ...
and it can be defined by :\mathrm(v,w)=(\nabla_v w)^\bot\,, where (\nabla_v w)^\bot denotes the orthogonal projection of
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
\nabla_v w onto the normal bundle. In
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, the curvature tensor of a
submanifold In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...
can be described by the following formula: :\langle R(u,v)w,z\rangle =\mathrm I\!\mathrm I(u,z)\mathrm I\!\mathrm I(v,w)-\mathrm I\!\mathrm I(u,w)\mathrm I\!\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 is a manifold embedded in a
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
then the curvature tensor of with induced metric can be expressed using the second fundamental form and , the curvature tensor of : :\langle R_N(u,v)w,z\rangle = \langle R_M(u,v)w,z\rangle+\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle\,.


See also

*
First fundamental form In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of . It permits the calculation of curvature and ...
*
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...
*
Gauss–Codazzi equations In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas) are fundamental formulas that link together the induced m ...
*
Shape operator In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...
* Third fundamental form *
Tautological one-form In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus pro ...


References

* * *


External links

* Steven Verpoort (2008
Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects
from
Katholieke Universiteit Leuven KU Leuven (Katholieke Universiteit Leuven) is a Catholic research university in the city of Leuven, Belgium. Founded in 1425, it is the oldest university in Belgium and the oldest university in the Low Countries. In addition to its main camp ...
. {{curvature Differential geometry Differential geometry of surfaces Riemannian geometry Curvature (mathematics) Tensors