note-math

Example Euclidean analysis of manifolds, various coordinates of the sphere ${\mathrm{𝕊}}^{𝑛}$

• Function graph coordinates, function equations $|𝑥{|}^2=1$ and implicit function theorem. e.g. $𝑦=\sqrt{1-{𝑥}^2}$ for ${\mathrm{𝕊}}^1\subset {\mathrm{ℝ}}^2$
• #link(<stereographic-projection>)[stereographic projection]
• Polar coordinates. Starting from trigonometric functions of ${\mathrm{𝕊}}^1$, construct new latitudes inductively
• Geodesic coordinates

Example Parametric curves and surfaces of ${\mathrm{ℝ}}^3$. analytic function $𝑓:{\mathrm{ℝ}}^2\to {\mathrm{ℝ}}^3$, $𝑑𝑓\ne 0$ ==> for local parameter, it's local analytic isomorphism

manifold_(tag) minimal structure to define manifold, family of coordinate cards covering $𝑀$ with the same dimension, transition functions using Euclidean or Minkowski or quadratic analysis

orientable_(tag) Orientable := can analytically define #link(<orientation>)[] in the tangent bundle

Equivalent to decomposition of $\text{Diff}$ to the ${\text{det}}^{-1}\left({\mathrm{ℝ}}_{<0}\right)\bigsqcup {\text{ det}}^{-1}\left({\mathrm{ℝ}}_{>0}\right)$

Equivalent to the existence of a coordinate cover, each transition function differentiation $𝑑𝑓\in \text{ SO}$

Example #link(<Mobius-strip>)[] Non-orientable

If the interior of a manifold with boundary is orientable, then the boundary is also orientable. Intuitively, the local of boundary has the same interior + the interior is orientable ==> local of boundary has the same orientation ==> the boundary orientation is determined

manifold-with-boundary_(tag) Manifold with boundaries. The coordinates can be the region enclosed by the $𝑛-1$-dimensional hyperplane, and the transformation function need to be able to derives the transformation function in the $𝑛-1$-dimensional subspace. Usually use almost everywhere analysis to deal with some singularities

metric-manifold_(tag) metric on manifold (Abbreviation metric) is to define metric in each tangent space, which is equivalent to choosing an orthonormal frame bundle on the manifold tangent bundle. For $\text{SO}\left(𝑝,𝑞\right)$ oritentable, we can choose $\text{SO}\left(𝑝,𝑞\right)$ orientable frame bundle

metric can be inherited from submanifold or quotient manifold of ${\mathrm{ℝ}}^{𝑝,𝑞}$

Example …

Although the manifold is defined using quadratic topology and differentials, there are still many different metrics. A well-behaved metric is #link(<Einstein-metric.typ>)[]

isometry_(tag) := diffeomorphism preserving metric $𝑔$. It is usually also assumed to preserve the orientation of the orientable manifold

Diffeomorphism acts on metric space, isometry is the #link(<isotropy>)[] of this group action

Metrics with different curvatures cannot be in the same orbit. In particular, zero-curvature and non-zero-curvature metrics cannot be in the same oribt

δ-isometry_(tag) alias Killing-field_(tag)

will be used for the momentum conservation flow on the manifold

Question dimension of δ-isometry and isometry group $\le \text{ dim }\left({\mathrm{ℝ}}^{𝑝,𝑞}⋊\text{ SO}\left(𝑝,𝑞\right)\right)$

Example some explicit construction of manifold

Quadratic manifold ${\mathrm{\mathbb{Q}}}^{𝑝,𝑞}\left(\pm {𝑟}^2\right)$

cf. ref-10 ref-11

group $\text{SO}\left(𝑝,𝑞\right),\text{U},\text{SU},\text{Sp},\text{SL},{𝐺}_2$. exp coordinate

Grassmannian-manifold_(tag) $\text{SO}\left(𝑛\right)$ act on $𝑘$ subspace $\frac{\text{SO}\left(𝑛\right)}{\text{SO}\left(𝑘\right)\times \text{SO}\left(𝑛-𝑘\right)}$ (orientable)

Stiefel-manifold_(tag) tautological frame bundle $\frac{\text{SO}\left(𝑛\right)}{\text{SO}\left(𝑛-𝑘\right)}$

tautological bundle

Generalized to the $𝑝,𝑞$ quadratic case

lens space

Continuous homeomorphism but not diffeomorphism. Example Various modifications of the quaternion $\mathrm{ℍ}$ version of #link(<Hopf-bundle>)[] give an example called exotic 7-shpere