If a compact Riemannian manifold has positive Ricci curvature then its fundamental group is finite.

If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line ( i. e. a geodesic which minimizes distance on each interval ) then it is isometric to a direct product of the real line and a complete ( n-1 )- dimensional Riemannian manifold which has nonnegative Ricci curvature.

If the Ricci curvature function Ric ( ξ, ξ ) is constant on the set of unit tangent vectors ξ, the Riemannian manifold is said to have constant Ricci curvature, or to be an Einstein manifold.

If you change the metric g by multiplying it by a conformal factor, the Ricci tensor of the new, conformally related metric is given by

If n 3, the trace-free Ricci tensor vanishes identically if and only if

If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the special linear group SL ( n, C ).

If we consider the metric tensor ( and the associated Ricci tensor ) to be functions of a variable which is usually called " time " ( but which may have nothing to do with any physical time ), then the Ricci flow may be defined by the geometric evolution equation

* If the manifold is Euclidean space, or more generally Ricci-flat, then Ricci flow leaves the metric unchanged.

* If the manifold is a sphere ( with the usual metric ) then Ricci flow collapses the manifold to a point in finite time.

If we adopt Cartesian coordinates with line element with corresponding wave operator on the flat background, or Minkowski spacetime, so that the line element of the curved spacetime is, then the Ricci scalar of this curved spacetime is just

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.

If T is a ( p, q )- tensor ( p for the contravariant vector and q for the covariant one ), then we define the divergence of T to be the ( p, q − 1 )- tensor

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

If the medium is anisotropic, the permittivity is a second rank tensor.

If one combines this with the symmetry of the stress – energy tensor, one can show that angular momentum is also conserved,

( If there is torsion, then the tensor is no longer symmetric.

If V and W are finite-dimensional, then the space of all linear transformations from W to V, denoted Hom ( W, V ), is generated by such outer products ; in fact, the rank of a matrix is the minimal number of such outer products needed to express it as a sum ( this is the tensor rank of a matrix ).

If it is considered as a tensor, the Kronecker tensor, it can be written

If we choose an orthonormal coordinate system () we can write the tensor in terms of components with respect to those base vectors as

If we are given the components of the strain tensor in an arbitrary orthonormal coordinate system, we can find the principal strains using an eigenvalue decomposition determined by solving the system of equations

If the elastic medium is visualised as a set of infinitesimal cubes in the unstrained state, after the medium is strained, an arbitrary strain tensor may not yield a situation in which the distorted cubes still fit together without overlapping.

If we have a field extension F / K, which is to say a bigger field F that contains K, then there is a natural way to construct an algebra over F from any algebra over K. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product.

An extension of the tensor field idea incorporates an extra line bundle L on M. If W is the tensor product bundle of V with L, then W is a bundle of vector spaces of just the same dimension as V. This allows one to define the concept of tensor density, a ' twisted ' type of tensor field.

If denotes an affine connection, then the curvature tensor is the tensor defined by

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

If is continuous in an open set Ω and the partial derivatives of ƒ with respect to x and y exist in Ω, and satisfies the Cauchy – Riemann equations throughout Ω, then ƒ is holomorphic ( and thus analytic ).

* If ƒ ( z ) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy – Riemann equations weakly, then ƒ agrees almost everywhere with an analytic function in Ω.

In computer science, a heap is a specialized tree-based data structure that satisfies the heap property: If A is a parent node of B then key ( A ) is ordered with respect to key ( B ) with the same ordering applying across the heap.

If the ranges of the morphisms of the inverse system of abelian groups ( A < sub > i </ sub >, f < sub > ij </ sub >) are stationary, that is, for every k there exists j ≥ k such that for all i ≥ j: one says that the system satisfies the Mittag-Leffler condition.

If a subspace satisfies a stronger condition that

If a model for a language moreover satisfies a particular sentence or theory ( set of sentences satisfying special conditions ), it is called a model of the sentence or theory.

If μ is not a positive measure, then N is μ-null if N is | μ |- null, where | μ | is the total variation of μ ; equivalently, if every measurable subset A of N satisfies μ ( A )

If the court satisfies itself that the defendant fully acknowledges the consequences of the plea agreement, and he / she was represented by the defense council, his / her will is expressed in full compliance with the legislative requirements without deception and coercion, also if there is enough body of doubtless evidence for the conviction and the agreement is reached on legitimate sentence-the court approves the plea agreement and renders guilty judgment.

If a function is harmonic ( that is, it satisfies Laplace's equation ) over a particular space, and is transformed via a conformal map to another space, the transformation is also harmonic.

If the string is stretched between two points where x = 0 and x = L and u denotes the amplitude of the displacement of the string, then u satisfies the one-dimensional wave equation in the region where 0 < x < L and t is unlimited.

If a delivery satisfies the criteria for both a No Ball and a Wide, the call and penalty of No Ball will take precedent.

If the heuristic h satisfies the additional condition for every edge x, y of the graph ( where d denotes the length of that edge ), then h is called monotone, or consistent.

If we now add another simultaneous equation to guarantee that we only perform this test when we are at a point that satisfies the

If the above integral is denoted by I ( b ), then the function I satisfies

In mathematics, and in particular linear algebra, a skew-symmetric ( or antisymmetric or antimetric ) matrix is a square matrix A whose transpose is also its negative ; that is, it satisfies the equation If the entry in the and is a < sub > ij </ sub >, i. e. then the skew symmetric condition is For example, the following matrix is skew-symmetric:

If is a basis, then the dual basis satisfies

If this trading strategy is followed, and any cash held is assumed to grow at the risk free rate r, then the total value V of this portfolio satisfies the SDE

* If H is a subgroup of G, the index of the normal core of H satisfies the following inequality:

If κ is measurable and p ∈ V < sub > κ </ sub > and M ( the ultrapower of V ) satisfies ψ ( κ, p ), then the set of α < κ such that V satisfies ψ ( α, p ) is stationary in κ ( actually a set of measure 1 ).

If we write in the form then the map satisfies

If a relief pitcher satisfies all of the criteria for a save, except he does not finish the game, he will often be credited with a hold ( which is not an officially recognized statistic by Major League Baseball ).

If is a fibration with fiber F, with the base B path-connected, and the fibration is orientable over a field K, then the Euler characteristic with coefficients in the field K satisfies the product property: