(flow, infinitesimal generator, integral curve, complete vector field)
Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is a flow domain. For each p ∈ M the map Dp → M is the unique maximal integral curve of V starting at p.
A global flow is one whose flow domain is all of R × M. Global flows define smooth actions of R on M. A vector field is complete if it generates a global flow. Every vector field on a compact manifold is complete.
(geodesic, exponential map, injectivity radius)
The exponential map
is defined as exp(X) = γ(1) where γ : I → M is the unique geodesic passing through p at 0 and whose tangent vector at 0 is X. Here I is the maximal open interval of R for which the geodesic is defined.
Let M be a pseudo-Riemannian manifold (or any manifold with an affine connection) and let p be a point in M. Then for every V in TpM there exists a unique geodesic γ : I → M for which γ(0) = p and . Let Dp be the subset of TpM for which 1 lies in I.
(exponential map, infinitesimal generator, one-parameter group)
Every left-invariant vector field on a Lie group is complete. The integral curve starting at the identity is a one-parameter subgroup of G. There are one-to-one correspondences
Let G be a Lie group and g its Lie algebra. The exponential map is a map exp : g → G given by exp(X) = γ(1) where γ is the integral curve starting at the identity in G generated by X.
The conformal group of a Riemannian manifold (M, g) is the group of conformal transformations of M. A conformal transformation is a diffeomorphism of M which locally rescales the metric by a positive function on M. Specifically, the conformal group is given by
For example, the conformal group of the n-sphere is isomorphic to the Lorentz group SO(n+1,1).
Let R∞ denote the space of all terminating sequences of real numbers (i.e. sequences with only a finite number of nonzero terms).
R∞ is subspace of the infinite product space . The topology is the initial topology with respect to the coordinate maps
i.e. the coarsest topology for which these maps are continuous.
R∞ is a real inner product space with the standard dot product (and hence a normed vector space and a metric space). It is not a Hilbert space or Banach space as the metric is not complete.
Properties:
R∞ is a CW complex with the associated topology.
R∞ is the direct limit limn→∞ Rn. The topology is the final topology with respect to the inclusions
i.e. the finest topology for which these maps are continuous.
less equal (≤) | inclusion | |
join (∨) | union (∪) | OR |
meet (∧) | intersection(∩) | AND |
complement (¬) | complement | NOT |
bottom (0) | empty set (∅) | |
top (1) | universe | |
plus (+) | symmetric difference | XOR |
See also:
Identities in special classes
left alternative | ||
right alternative | ||
flexible |
A loop Q is
Moufang loops are both left and right Bol. Moreover, any loop which is both left and right Bol is Moufang. However, the Bol identities by themselves are strictly weaker than the Moufang identities.
Every left Bol loop is left alternative and satisfies the left inverse property. In fact, a loop is left Bol iff every loop isotope of it satisfies the left inverse property. It follows that every isotope of a left Bol loop is left Bol. Dual statements apply to right Bol loops.
A left Bol loop is a Moufang loop if it satifies any of: the flexible law, the right alternative law, or the right inverse property. Similarly, for right Bol loops. Any commutative Bol loop is Moufang.
There are two variants of the Cayley-Dickson construction depending on whether you prefer to write the new imaginary unit on the left or the right:
and
where
To "derive" these one uses the manipulations:
The latter three manipulations assume that x, y, and ℓ obey the Moufang identities (at least when ℓ is the doubled variable).
The p-adic solenoid is the topological group defined as the inverse limit
where each Ti is a copy of the circle group T and qi is the map that takes the pth power of its argument (and therefore wraps Ti+1 around Ti p times). Explicitly, the elements of Sp can be described as infinite sequences of elements from T with each coordinate being the pth power of the next coordinate:
The topology of Sp is the initial topology with respect to the projection maps.
By projecting onto the first coordinate we get a homomorphism from Sp to T. The kernel of this homomorphism is isomorphic to the group of p-adic integers Zp. We then have a short exact sequence of togological groups:
Topologically, p-adic integers form a Cantor space so the solenoid can be described as a fiber bundle over the circle with a Cantor space fiber.
Every metric space is
Every CW complex is
Every manifold (assumed Hausdorff) is
Every paracompact manifold is the above plus
Every second countable manifold is the above plus
A division algebra is an algebra A over a field K for which the operators are invertible for each nonzero a ∈ A. We do not assume A to be unital, associative, or finite-dimensional.
(flow, infinitesimal generator, integral curve, complete vector field)
Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is a flow domain. For each p ∈ M the map Dp → M is the unique maximal integral curve of V starting at p.
A global flow is one whose flow domain is all of R × M. Global flows define smooth actions of R on M. A vector field is complete if it generates a global flow. Every vector field on a compact manifold is complete.
(geodesic, exponential map, injectivity radius)
The exponential map
is defined as exp(X) = γ(1) where γ : I → M is the unique geodesic passing through p at 0 and whose tangent vector at 0 is X. Here I is the maximal open interval of R for which the geodesic is defined.
Let M be a pseudo-Riemannian manifold (or any manifold with an affine connection) and let p be a point in M. Then for every V in TpM there exists a unique geodesic γ : I → M for which γ(0) = p and . Let Dp be the subset of TpM for which 1 lies in I.
(exponential map, infinitesimal generator, one-parameter group)
Every left-invariant vector field on a Lie group is complete. The integral curve starting at the identity is a one-parameter subgroup of G. There are one-to-one correspondences
Let G be a Lie group and g its Lie algebra. The exponential map is a map exp : g → G given by exp(X) = γ(1) where γ is the integral curve starting at the identity in G generated by X.
The conformal group of a Riemannian manifold (M, g) is the group of conformal transformations of M. A conformal transformation is a diffeomorphism of M which locally rescales the metric by a positive function on M. Specifically, the conformal group is given by
For example, the conformal group of the n-sphere is isomorphic to the Lorentz group SO(n+1,1).
Let R∞ denote the space of all terminating sequences of real numbers (i.e. sequences with only a finite number of nonzero terms).
R∞ is subspace of the infinite product space . The topology is the initial topology with respect to the coordinate maps
i.e. the coarsest topology for which these maps are continuous.
R∞ is a real inner product space with the standard dot product (and hence a normed vector space and a metric space). It is not a Hilbert space or Banach space as the metric is not complete.
Properties:
R∞ is a CW complex with the associated topology.
R∞ is the direct limit limn→∞ Rn. The topology is the final topology with respect to the inclusions
i.e. the finest topology for which these maps are continuous.
less equal (≤) | inclusion | |
join (∨) | union (∪) | OR |
meet (∧) | intersection(∩) | AND |
complement (¬) | complement | NOT |
bottom (0) | empty set (∅) | |
top (1) | universe | |
plus (+) | symmetric difference | XOR |
See also:
Identities in special classes
left alternative | ||
right alternative | ||
flexible |
A loop Q is
Moufang loops are both left and right Bol. Moreover, any loop which is both left and right Bol is Moufang. However, the Bol identities by themselves are strictly weaker than the Moufang identities.
Every left Bol loop is left alternative and satisfies the left inverse property. In fact, a loop is left Bol iff every loop isotope of it satisfies the left inverse property. It follows that every isotope of a left Bol loop is left Bol. Dual statements apply to right Bol loops.
A left Bol loop is a Moufang loop if it satifies any of: the flexible law, the right alternative law, or the right inverse property. Similarly, for right Bol loops. Any commutative Bol loop is Moufang.
There are two variants of the Cayley-Dickson construction depending on whether you prefer to write the new imaginary unit on the left or the right:
and
where
To "derive" these one uses the manipulations:
The latter three manipulations assume that x, y, and ℓ obey the Moufang identities (at least when ℓ is the doubled variable).
The p-adic solenoid is the topological group defined as the inverse limit
where each Ti is a copy of the circle group T and qi is the map that takes the pth power of its argument (and therefore wraps Ti+1 around Ti p times). Explicitly, the elements of Sp can be described as infinite sequences of elements from T with each coordinate being the pth power of the next coordinate:
The topology of Sp is the initial topology with respect to the projection maps.
By projecting onto the first coordinate we get a homomorphism from Sp to T. The kernel of this homomorphism is isomorphic to the group of p-adic integers Zp. We then have a short exact sequence of togological groups:
Topologically, p-adic integers form a Cantor space so the solenoid can be described as a fiber bundle over the circle with a Cantor space fiber.
Every metric space is
Every CW complex is
Every manifold (assumed Hausdorff) is
Every paracompact manifold is the above plus
Every second countable manifold is the above plus
A division algebra is an algebra A over a field K for which the operators are invertible for each nonzero a ∈ A. We do not assume A to be unital, associative, or finite-dimensional.