User:Fropuff/Notes

Covers

• The set of all open covers of a topological space is a directed set under refinement (refinement is only a preorder, not a partial order).
• Every manifold has a good cover, every compact manifold has a finite good cover. Furthermore, every open cover has a refinement which is a good cover, i.e. the set of good covers of a manifold is cofinal in the set of all open covers.

Exponential map and vector flows

• exponential map
  • matrix exponential
  • exponential function
• infinitesimal generator (→ Lie group)
• integral curve (→ vector field)
• one-parameter subgroup
• flow (geometry)
  • geodesic flow (→ glossary)
  • Ricci flow
• injectivity radius (→ glossary)

Differential topology

(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 D_p → 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.

Riemannian geometry

(geodesic, exponential map, injectivity radius)

The exponential map

exp : T_pM → M

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 T_pM there exists a unique geodesic γ : I → M for which γ(0) = p and ${\dot {\gamma }}(0)=V$. Let D_p be the subset of T_pM for which 1 lies in I.

Lie group theory

(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

{one-parameter subgroups of G} ⇔ {left-invariant vector fields on G} ⇔ g = T_eG.

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 exponential map is smooth.
• For a fixed X, the map t |-> exp(tX) is the one-parameter subgroup of G generated by X.
• The exponetial map restricts to a diffeomorphism form some neighborhood of 0 in g to a neighborhood of e in G.
• The image of the exponential map always lies in the connected component of the identity in G.

Conformal group

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

$\mathrm {Conf} (M)=\{f\in \mathrm {Diff} (M):f^{*}g=e^{\Lambda }g{\mbox{ for some }}\Lambda \in C^{\infty }(M)\}$

For example, the conformal group of the n-sphere is isomorphic to the Lorentz group SO(n+1,1).

Infinite coordinate space

Let R^∞ denote the space of all terminating sequences of real numbers (i.e. sequences with only a finite number of nonzero terms).

Initial topology

R^∞ is subspace of the infinite product space $\prod _{i=1}^{\infty }\mathbb {R}$. The topology is the initial topology with respect to the coordinate maps

$p_{i}\colon \mathbb {R} ^{\infty }\to \mathbb {R}$

i.e. the coarsest topology for which these maps are continuous.

Metric topology

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:

• metrizable and thus
  • perfectly normal Hausdorff
  • first countable
  • paracompact
  • compactly generated
• not locally compact
• the unit sphere S^∞ is not compact

CW topology

R^∞ is a CW complex with the associated topology.

Final topology

R^∞ is the direct limit lim_n→∞ Rⁿ. The topology is the final topology with respect to the inclusions

$\phi _{i}\colon \mathbb {R} ^{i}\to \mathbb {R} ^{\infty }$

i.e. the finest topology for which these maps are continuous.

Boolean algebras

less equal (≤)	inclusion
join (∨)	union (∪)	OR
meet (∧)	intersection(∩)	AND
complement (¬)	complement	NOT
bottom (0)	empty set (∅)
top (1)	universe
plus (+)	symmetric difference	XOR

Articles dealing with rotations

• rotation
• coordinate rotation
• coordinate rotations and reflections
• rotation matrix
• rotation group
• rotation operator
• Euler's rotation theorem
• Rodrigues' rotation formula
• charts on SO(3)
• quaternions and spatial rotations
• Euler-Rodrigues parameters
• Euler angles
• improper rotation

See also:

• orthogonal group
• spin group
• Lorentz group
• Clifford algebra

Quasigroups and Loops

Identities in special classes

left alternative	$x(xy)=(xx)y$	$L_{x}L_{x}=L_{x^{2}}$
right alternative	$(yx)x=y(xx)$	$R_{x}R_{x}=R_{x^{2}}$
flexible	$x(yx)=(xy)x$	$L_{x}R_{x}=R_{x}L_{x}$

Bol loops

A loop Q is

• a left Bol loop if $z(x(zy))=(z(xz))y$ for all x, y, and z in Q
• a right Bol loop if $((xz)y)z=x((zy)z)$ for all x, y, and z in Q

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.

Cayley-Dickson construction

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:

$(a+b\ell )(c+d\ell )=(ac+\lambda {\bar {d}}b)+(da+b{\bar {c}})\ell$

and

$(a+\ell b)(c+\ell d)=(ac+\lambda d{\bar {b}})+\ell ({\bar {a}}d+cb)$

where

$\lambda =\ell ^{2}\in K^{\times }$

To "derive" these one uses the manipulations:

• ${\overline {xy}}={\bar {y}}{\bar {x}}$
• $\ell x={\bar {x}}\ell$
• $(\ell x)(y\ell )=(\ell (xy))\ell =\ell (\ell ({\bar {y}}{\bar {x}})=\lambda ({\bar {y}}{\bar {x}})$
• $x(\ell y)=x({\bar {y}}\ell )=\ell ({\bar {x}}y)=({\bar {y}}x)\ell$
• $(x\ell )y=(\ell {\bar {x}})y=(x{\bar {y}})\ell =\ell (y{\bar {x}})$

The latter three manipulations assume that x, y, and ℓ obey the Moufang identities (at least when ℓ is the doubled variable).

Solenoid group

The p-adic solenoid is the topological group defined as the inverse limit

$S_{p}=\varprojlim ({\mathbb {T} }_{i},q_{i}).$

where each T_i is a copy of the circle group T and q_i is the map that takes the pth power of its argument (and therefore wraps T_i+1 around T_i p times). Explicitly, the elements of S_{p can be described as infinite sequences of elements from T with each coordinate being the pth power of the next coordinate:}

$S_{p}=\left\{(a_{i})_{i\in \mathbb {N} }:a_{i}\in \mathbb {T} {\mbox{ and }}a_{i}=a_{i+1}^{p}\right\}.$

The topology of S_p is the initial topology with respect to the projection maps.

By projecting onto the first coordinate we get a homomorphism from S_p to T. The kernel of this homomorphism is isomorphic to the group of p-adic integers Z_p. We then have a short exact sequence of togological groups:

$0\to \mathbb {Z} _{p}\to S_{p}\to \mathbb {T} \to 0.$

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.

Topological properties

Metric spaces

Every metric space is

• perfectly normal Hausdorff
• first countable and therefore
  • compactly generated
• paracompact

CW complexes

Every CW complex is

• normal Hausdorff
• locally contractible and therefore
  • locally simply connected
  • locally path connected
  • locally connected
• compactly generated
• paracompact

Manifolds

Every manifold (assumed Hausdorff) is

• first countable
• locally compact and therefore
  • compactly generated
• Tychonoff (completely regular Hausdorff)
• locally contractible and therefore
  • locally simply connected
  • locally path connected
  • locally connected.

Every paracompact manifold is the above plus

• paracompact
• metrizable and therefore
  • perfectly normal Hausdorff

Every second countable manifold is the above plus

• second countable and therefore
  • separable space
  • Lindelöf space

Articles dealing with curvature

• curvature
• curvature tensor → Riemann curvature tensor
• curvature form
• curvature of Riemannian manifolds
• sectional curvature
• Ricci curvature
• Weyl curvature → Weyl tensor
• scalar curvature
• constant curvature
• Gaussian curvature
• principal curvature
• mean curvature
• differential geometry of curves
• torsion tensor

Division algebras

A division algebra is an algebra A over a field K for which the operators $L_{a},R_{a}$ are invertible for each nonzero a ∈ A. We do not assume A to be unital, associative, or finite-dimensional.

• Every finite-dimensional associative division algebra over the reals is isomorphic to R, C, or H. (Frobenius theorem, 1878)
• Every finite-dimensional alternative division algebra over the reals is isomorphic to R, C, H, or O. (Zorn (?), 1930)
• Every normed division algebra over the reals is isomorphic to R, C, H, or O. (Hurwitz's theorem, 1898)