Second fundamental form

extrinsic curvaturesecondshape tensor
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").wikipedia
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Quadratic form

quadratic formssignaturequadratic space
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).

Parametric surface

Curvature of parametric surfacesparameterizeparametric
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.

Principal curvature

principal curvaturesprincipal directionsprincipal radii of curvature
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.

First fundamental form

firstfirst quadratic forms
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.

Gauss–Codazzi equations

Gauss-Codazzi equationsGauss–Codazzi equationGauss–Codazzi equations (relativity)
: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.

Gaussian curvature

Gauss curvaturecurvatureLiebmann's theorem
Equivalently, the determinant of the second fundamental form of a surface in

Theorema Egregium

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

Differential geometry

differentialdifferential geometerdifferential geometry and topology
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").

Tangent space

tangent planetangenttangent vector
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").

Euclidean space

EuclideanspaceEuclidean vector space
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").

Hypersurface

complex hypersurfaceprojective hypersurfacesurface
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.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
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.

Differentiable function

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

Tangent

tangent linetangentialtangents
is tangent to the surface at the origin.

Partial derivative

partial derivativespartial differentiationpartial differential
and its partial derivatives with respect to

Taylor series

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

Vector-valued function

vector functionvector-valued functionsvector
is a smooth vector-valued function of two variables.

Cross product

vector cross productvector productcross-product
Equivalently, the cross product

Dot product

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

Einstein notation

Einstein summation conventionsummation conventionEinstein summation notation
The equation above uses the Einstein summation convention.

Vector-valued differential form

tensorial formvector-valued formwith values in
regarded as a vector-valued differential form, and the brackets denote the metric tensor of Euclidean space.