In
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
and
pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi Formulas) are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of (or immersion into) a
Riemannian or
pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
.
The equations were originally discovered in the context of surfaces in three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. In this context, the first equation, often called the Gauss equation (after its discoverer
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
), says that the
Gauss curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
of the surface, at any given point, is dictated by the derivatives of the Gauss map at that point, as encoded by the
second fundamental form
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 \mathrm (read "two"). Together with the first fundame ...
. The second equation, called the Codazzi equation or Codazzi-Mainardi equation, states that the
covariant derivative of the second fundamental form is fully symmetric. It is named for
Gaspare Mainardi (1856) and
Delfino Codazzi (1868–1869), who independently derived the result, although it was discovered earlier by
Karl Mikhailovich Peterson.
Formal statement
Let
be an ''n''-dimensional embedded submanifold of a Riemannian manifold ''P'' of dimension
. There is a natural inclusion of the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of ''M'' into that of ''P'' by the
pushforward
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things.
* Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
, and the
cokernel
The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of .
Cokernels are dual to the kernels of category theory, hence the nam ...
is 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 m ...
of ''M'':
:
The metric splits this
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
, and so
:
Relative to this splitting, the
Levi-Civita connection of ''P'' decomposes into tangential and normal components. For each
and vector field ''Y'' on ''M'',
:
Let
:
The Gauss formula now asserts that
is the
Levi-Civita connection for ''M'', and
is a ''symmetric''
vector-valued form with values in the normal bundle. It is often referred to as the
second fundamental form
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 \mathrm (read "two"). Together with the first fundame ...
.
An immediate corollary is the Gauss equation for the curvature tensor. For
,
:
where
is the
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
of ''P'' and ''R'' is that of ''M''.
The
Weingarten equation is an analog of the Gauss formula for a connection in the normal bundle. Let
and
a normal vector field. Then decompose the ambient covariant derivative of
along ''X'' into tangential and normal components:
:
Then
# ''Weingarten's equation'':
# ''D''
X is a
metric connection in the normal bundle.
There are thus a pair of connections: ∇, defined on the tangent bundle of ''M''; and ''D'', defined on the normal bundle of ''M''. These combine to form a connection on any tensor product of copies of T''M'' and T
⊥''M''. In particular, they defined the covariant derivative of
:
:
The Codazzi–Mainardi equation is
:
Since every
immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
is, in particular, a local embedding, the above formulas also hold for immersions.
Gauss–Codazzi equations in classical differential geometry
Statement of classical equations
In classical
differential geometry of surfaces, the Codazzi–Mainardi equations are expressed via the
second fundamental form
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 \mathrm (read "two"). Together with the first fundame ...
(''L'', ''M'', ''N''):
:
:
The Gauss formula, depending on how one chooses to define the Gaussian curvature, may be a
tautology. It can be stated as
:
where (''e'', ''f'', ''g'') are the components of the first fundamental form.
Derivation of classical equations
Consider a
parametric surface in Euclidean 3-space,
:
where the three component functions depend smoothly on ordered pairs (''u'',''v'') in some open domain ''U'' in the ''uv''-plane. Assume that this surface is regular, meaning that the vectors r
''u'' and r
''v'' are
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. Complete this to a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
, by selecting a unit vector n normal to the surface. It is possible to express the second partial derivatives of r (vectors of
) with the
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distance ...
and the elements of the second fundamental form. We choose the first two components of the basis as they are intrinsic to the surface and intend to prove intrinsic property of the
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
. The last term in the basis is extrinsic.
:
:
:
Clairaut's theorem
Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatis ...
states that partial derivatives commute:
:
If we differentiate r
uu with respect to ''v'' and r
uv with respect to ''u'', we get:
:
Now substitute the above expressions for the second derivatives and equate the coefficients of n:
:
Rearranging this equation gives the first Codazzi–Mainardi equation.
The second equation may be derived similarly.
Mean curvature
Let ''M'' be a smooth ''m''-dimensional manifold immersed in the (''m'' + ''k'')-dimensional smooth manifold ''P''. Let
be a local orthonormal frame of vector fields normal to ''M''. Then we can write,
:
If, now,
is a local orthonormal frame (of tangent vector fields) on the same open subset of ''M'', then we can define the
mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
s of the immersion by
:
In particular, if ''M'' is a hypersurface of ''P'', i.e.
, then there is only one mean curvature to speak of. The immersion is called
minimal if all the
are identically zero.
Observe that the mean curvature is a trace, or average, of the second fundamental form, for any given component. Sometimes mean curvature is defined by multiplying the sum on the right-hand side by
.
We can now write the Gauss–Codazzi equations as
:
Contracting the
components gives us
:
When ''M'' is a hypersurface, this simplifies to
:
where
and
. In that case, one more contraction yields,
:
where
and
are the scalar curvatures of ''P'' and ''M'' respectively, and
:
If
, the scalar curvature equation might be more complicated.
We can already use these equations to draw some conclusions. For example, any minimal immersion
into the round sphere
must be of the form
:
where
runs from 1 to
and
:
is the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
on ''M'', and
is a positive constant.
See also
*
Darboux frame In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a s ...
Notes
References
Historical references
*
*
* ("General Discussions about Curved Surfaces")
*
*
*
*.
Textbooks
* do Carmo, Manfredo P. ''Differential geometry of curves & surfaces.'' Revised & updated second edition. Dover Publications, Inc., Mineola, NY, 2016. xvi+510 pp.
* do Carmo, Manfredo Perdigão. ''Riemannian geometry.'' Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. xiv+300 pp.
* Kobayashi, Shoshichi; Nomizu, Katsumi. ''Foundations of differential geometry. Vol. II.'' Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1969 xv+470 pp.
* O'Neill, Barrett. ''Semi-Riemannian geometry. With applications to relativity.'' Pure and Applied Mathematics, 103. Academic Press, Inc.
arcourt Brace Jovanovich, Publishers New York, 1983. xiii+468 pp.
*V. A. Toponogov. ''Differential geometry of curves and surfaces. A concise guide''. Birkhauser Boston, Inc., Boston, MA, 2006. xiv+206 pp. ; .''
Articles
*
* Simons, James. ''Minimal varieties in riemannian manifolds.'' Ann. of Math. (2) 88 (1968), 62–105.
External links
Peterson–Mainardi–Codazzi Equations – from Wolfram MathWorldPeterson–Codazzi Equations
{{DEFAULTSORT:Gauss-Codazzi equations
Differential geometry of surfaces
Riemannian geometry
Curvature (mathematics)
Surfaces