Example Euclidean analysis of manifolds, various coordinates of the sphere
- Function graph coordinates, function equations and implicit function theorem. e.g. for
- stereographic projection
- Polar coordinates. Starting from trigonometric functions of , construct new latitudes inductively
- Geodesic coordinates
Example Parametric curves and surfaces of . analytic function , ==> for local parameter, it’s local analytic isomorphism
[manifold] Many things are invariant under local analytically isomorphism, so manifold can be defined to be the generalization of local analytically isomorphism (or local diffeomorphism), family of coordinate cards covering with the same dimension, transition functions using Euclidean or Minkowski or quadratic analysis
[orientable] Orientable := can analytically define orientation in the tangent bundle
Equivalent to decomposition of to the
Equivalent to the existence of a coordinate cover, each transition function differentiation
Example 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] Manifold with boundaries. The coordinates can be the region enclosed by the -dimensional hyperplane, and the transformation function need to be able to derives the transformation function in the -dimensional subspace.
[metric-manifold] 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 oritentable, we can choose orientable frame bundle
[submanifold] identity embedding is manifold homomorphism. equivalently, local diffeomorphism of locally flatten submanifold
[quotient-manifold] …
metric can be inherited from submanifold or quotient manifold of
Example …
Although the manifold is defined using quadratic topology and differentials, there are still many different metrics. A well-behaved metric is Einstein-metric.typ
[isometry] := diffeomorphism preserving metric . It is usually also assumed to preserve the orientation of the orientable manifold
Diffeomorphism acts on metric space, isometry is the 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] alias [Killing-field]
will be used for the momentum conservation flow on the manifold
Question dimension of δ-isometry and isometry group
Example some explicit construction of manifold
Quadratic manifold
group . exp coordinate
[Grassmannian-manifold] act on subspace (orientable)
[Stiefel-manifold] tautological frame bundle
tautological bundle
Generalized to the quadratic case
lens space
Continuous homeomorphism but not diffeomorphism. Example Various modifications of the quaternion version of Hopf-bundle give an example called exotic 7-shpere