# Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket = XY − YX, because the Lie bracket of any two derivations is a derivation.

# If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.

# We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations L < sub > g </ sub >( h ) = gh.

# Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold.

# A Riemann surface X is a complex manifold of complex dimension one.

# A Riemann surface is an oriented manifold of ( real ) dimension two – a two-sided surface – together with a conformal structure.

# 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 geodesic flow of any compact Riemannian manifold with negative sectional curvature is ergodic.

# If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT ( k ) space.

# The isometry group of a compact Riemannian manifold with negative Ricci curvature is discrete.

# Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.

# If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n ( n-1 ).

# As the cultural component of human empowerment, emancipative values are highly consequential in manifold ways.

# 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 Bishop – Gromov inequality states that if a complete m-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a ball is smaller or equal to the volume of a ball of the same radius in Euclidean m-space.

# 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.

Bottom left and right: 2D recoveries of the manifold respectively using the Nonlinear dimensionality reduction # Locally-linear_embedding | LLE and Nonlinear dimensionality reduction # Hessian_LLE | Hessian LLE algorithms as implemented by the Modular Data Processing toolkit.

# Nash embedding theorems also called fundamental theorems of Riemannian geometry.

# REDIRECT Glossary of Riemannian and metric geometry # I

# Pontryagin numbers of closed Riemannian manifold ( as well as Pontryagin classes ) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.

# Riemannian geometry

# Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 ( real ) dimensions.

* Glossary of Riemannian and metric geometry # R for its meaning in Riemannian geometry.

# REDIRECT Embedding # Riemannian geometry

# REDIRECT Riemannian manifold

# Managed-the process is quantitatively managed in accordance with agreed-upon metrics.

# pre-defined “ earning rules ” ( also called metrics ) to quantify the accomplishment of work, called Earned Value ( EV ) or Budgeted Cost of Work Performed ( BCWP ).

# Measurement – program that creates a hierarchy of performance metrics ( see also Metrics Reference Model ) and benchmarking that informs business leaders about progress towards business goals ( business process management ).

# Give metrics on data quality, including whether the data conforms to particular standards or patterns

# Economic Wellness: Indicated via direct survey and statistical measurement of economic metrics such as consumer debt, average income to consumer price index ratio and income distribution

# Environmental Wellness: Indicated via direct survey and statistical measurement of environmental metrics such as pollution, noise and traffic

# Physical Wellness: Indicated via statistical measurement of physical health metrics such as severe illnesses

# Mental Wellness: Indicated via direct survey and statistical measurement of mental health metrics such as usage of antidepressants and rise or decline of psychotherapy patients

# Workplace Wellness: Indicated via direct survey and statistical measurement of labor metrics such as jobless claims, job change, workplace complaints and lawsuits

# Social Wellness: Indicated via direct survey and statistical measurement of social metrics such as discrimination, safety, divorce rates, complaints of domestic conflicts and family lawsuits, public lawsuits, crime rates

# Political Wellness: Indicated via direct survey and statistical measurement of political metrics such as the quality of local democracy, individual freedom, and foreign conflicts.

# Integrating Risks: This includes the aggregation of all risk distributions, reflecting correlations and portfolio effects, and the formulation of the results in terms of impact on the organization ’ s key performance metrics.

This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three dimensional Riemannian manifold.

The divergence can be defined on any manifold of dimension n with a volume form ( or density ) e. g. a Riemannian or Lorentzian manifold.

On a Riemannian or Lorentzian manifold the divergence with respect to the metric volume form

For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold — that is, one can smoothly " flatten out " certain manifolds, but it might require distorting the space and affecting the curvature or volume.

In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat.

However, if one additionally prescribes a smoothly varying inner product on the manifold's tangent spaces, then the result is what is called a Riemannian manifold.

Put differently, a Riemannian manifold is a space constructed by deforming and patching together Euclidean spaces.

The simplest Riemannian manifold, consisting of R < sup > n </ sup > with a constant inner product, is essentially identical to Euclidean n-space itself.

The base space of Kaluza – Klein theory need not be four-dimensional space-time ; it can be any ( pseudo -) Riemannian manifold, or even a supersymmetric manifold or orbifold or even a noncommutative space.

Let M be a ( pseudo -) Riemannian manifold, which may be taken as the spacetime of general relativity.

On a " global " level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure.

* Riemannian manifold

Conformal maps can be defined between domains in higher dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold.

In Riemannian geometry, two Riemannian metrics and on smooth manifold are called conformally equivalent if for some positive function on.

One can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics.

The Nash embedding theorems ( or imbedding theorems ), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space.

Let ( M, g ) be a Riemannian manifold and ƒ: M < sup > m </ sup > → R < sup > n </ sup > a short C < sup >∞</ sup >- embedding ( or immersion ) into Euclidean space R < sup > n </ sup >, where n ≥ m + 1.

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.

The technical statement is as follows: if M is a given m-dimensional Riemannian manifold ( analytic or of class C < sup > k </ sup >, 3 ≤ k ≤ ∞), then there exists a number n ( with n ≤ m ( 3m + 11 )/ 2 if M is a compact manifold, or n ≤ m ( m + 1 )( 3m + 11 )/ 2 if M is a non-compact manifold ) and an injective map ƒ: M → R < sup > n </ sup > ( also analytic or of class C < sup > k </ sup >) such that for every point p of M, the derivative dƒ < sub > p </ sub > is a linear map from the tangent space T < sub > p </ sub > M to R < sup > n </ sup > which is compatible with the given inner product on T < sub > p </ sub > M and the standard dot product of R < sup > n </ sup > in the following sense:

There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space ( usually a Euclidean space ) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in a Riemannian manifold.