* Fundamental theorem of arithmetic

* 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (" New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors ( i. e., polynomials ) of the first or second degree ")

Fundamental theorem of arithmetic.

simple: Fundamental theorem of arithmetic

# REDIRECT Fundamental theorem of arithmetic

* Fundamental theorem of analog circuits: a minimum level of power must be dissipated to maintain a level of SNR

This fact has led some to remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra.

# REDIRECT Fundamental theorem on homomorphisms

Interest in prime elements comes from the Fundamental theorem of arithmetic, which asserts that each integer can be written in essentially only one way as 1 or − 1 multiplied by a product of positive prime numbers.

* Fundamental theorem of software engineering

Fundamental theorem of poker

This is the Fundamental theorem of algebra.

* Fundamental theorem of calculus

* Fundamental theorem of arithmetic –

The first Fundamental theorem is a simple consequence

The Second Fundamental theorem says that for every k distinct values a < sub > j </ sub > on the Riemann sphere, we have

This theorem is called the Second Fundamental Theorem of Nevanlinna Theory, and it allows to give an upper bound for the characteristic function in terms of N ( r, a ).

For example, if f is a transcendental entire function, using the Second Fundamental theorem with k = 3 and a < sub > 3 </ sub > = ∞, we obtain that f takes every value infintiely often, with at most two exceptions,

* Fundamental theorem of vector analysis

* Fundamental theorem of arbitrage-free pricing

* Fundamental theorem of vector analysis

By the Fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography.

# REDIRECT Fundamental theorem of asset pricing

See Fundamental theorem of arbitrage-free pricing.

Fundamental groups also appear in algebraic geometry and are the main topic of Alexander Grothendieck's first Séminaire de géométrie algébrique ( SGA1 ).

* Fundamental theorem of projective geometry

* Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Fundamental algebraic geometry.

In particular, as follows from the Whitney embedding theorem, any m-dimensional Riemannian manifold admits an isometric C < sup > 1 </ sup >- embedding into an arbitrarily small neighborhood in 2m-dimensional Euclidean space.

Generalizations of the Gauss – Bonnet theorem to n-dimensional Riemannian manifolds were found in the 1940s, by Allendoerfer, Weil, and Chern ; see generalized Gauss – Bonnet theorem and Chern – Weil homomorphism.

In fact, as follows from the Nash embedding theorem, all Riemannian manifolds can be realized this way.

# Gauss – Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ ( M ) where χ ( M ) denotes the Euler characteristic of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem.

# The Cartan – Hadamard theorem states that a complete simply connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space R ^ n with n = dim M via the exponential map at any point.

The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties.

# Myers ' theorem states that if the Ricci curvature is bounded from below on a complete Riemannian manifold by, then the manifold has diameter, with equality only if the manifold is isometric to a sphere of a constant curvature k. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental group.

# The Cheeger-Gromoll splitting theorem states that if a complete Riemannian manifold with contains a line, meaning a geodesic γ such that for all, then it is isometric to a product space.

In particular, the fundamental theorem of Riemannian geometry is true of pseudo-Riemannian manifolds as well.

The uniformization theorem implies a similar result for arbitrary connected second countable surfaces: they can be given Riemannian metrics of constant curvature.

The Bonnet – Myers theorem states that a complete Riemannian manifold which has Ricci curvature everywhere greater than a certain positive constant must be compact.

A recent deep theorem states that M as a Riemannian manifold can be recovered from this data.

In mathematics, the generalized Gauss – Bonnet theorem ( also called Chern – Gauss – Bonnet theorem ) presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature.

Another simple and very useful result in Riemannian geometry is Gromov's compactness theorem, which states that

In 1952 Georges de Rham proved the de Rham decomposition theorem, a principle for splitting a Riemannian manifold into a Cartesian product of Riemannian manifolds by splitting the tangent bundle into irreducible spaces under the action of the local holonomy groups.

On the way to his classification of Riemannian holonomy groups, Berger developed two criteria that must be satisfied by the Lie algebra of the holonomy group of a torsion-free affine connection which is not locally symmetric: one of them, known as Berger's first criterion, is a consequence of the Ambrose – Singer theorem, that the curvature generates the holonomy algebra ; the other, known as Berger's second criterion, comes from the requirement that the connection should not be locally symmetric.

In mathematics, the Hopf – Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds.