# 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.

### Third fundamental form

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

### 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**

### 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.

### Carl Friedrich Gauss

**GaussCarl GaussCarl Friedrich Gauß**

was introduced and studied by Gauss.

### 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.

### Pushforward (differential)

**pushforwarddifferentialderivative**

the differential of

### 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.