Submanifold

Differentiable Functions

Let F be a function from a manifold M1 of dimension d1 into another manifold M2 of dimension d2. Let x be a point of M1. Choosing charts φ1 and φ2 around x and F(x), respectively, the function F around x can be “read through the charts”, yielding the function
- ˆF = φ2 ◦ F ◦ φ1^−1 : R^d1 → R^d2
- called a coordinate representation of F.
We say that F is differentiable or smooth at x if ˆF is of class C^∞ at φ1(x).
- C^∞ indicates that it is a C-infinity function, or infinitely differentiable.
- The operation ∘ is function composition, which takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)).
A (smooth) diﬀeomorphism F : M1 → M2 is a bijection such that F and its inverse F^−1 are both smooth.

The concepts of immersion and submersion will make it possible to deﬁne submanifolds and quotient manifolds in a concise way.
Let F : M1 → M2 be a differentiable function from a manifold M1 of dimension d1 into a manifold M2 of dimension d2. Let ˆF be the coordinate representation of F around x, and
- DˆF (φ1(x)) [·] : R^d1 → R^d2
- denotes the differential of ˆF at φ1(x).
The function F is called an immersion if DˆF (φ1(x)) is an injective (one-to-one) function at each point of M1. (Note that function ˆF itself need not be injective, only its differential must be.) The function F is called an submersion if DˆF (φ1(x)) is an surjective (onto) function at each point of M1.
Another equivalent way to define immersion and submersion involves the concept of rank.
- Given a point x of M1, the rank of F at x is the dimension of the range of DˆF (φ1(x)).
- The function F is called an immersion if its rank is equal to d1 at each point of its domain (hence d1 ≤ d2). If its rank is equal to d2 at each point of its domain (hence d1 ≥ d2), then it is called a submersion.

Let (M, A+) and (N, B+) be manifolds such that N ⊂ M.
The manifold (N, B+) is called an immersed submanifold of (M, A+) if the inclusion map i : N → M : x → x is an immersion.
- Note that the map itself does not necessarily have to be injective, allowing for the possibility of self-intersection within the larger manifold. If we require the map itself to be injective, this can prevent it from self-intersection, but this condition is not sufficient for topological embedding.
For instance, a figure-eight curve in the plane is an immersed 1-manifold in the 2-manifold of the plane, but not an embedded submanifold because it intersects itself. This self-intersection violates the topological embedding requirement to be defined in the embedded submanifolds.

Let (N, B+) be an immersed submanifold of (M, A+).
The manifold (N, B+) is called an embedded submanifold (or a regular submanifold, or simply a submanifold) of (M, A+) if the inclusion map is also a topological embedding.
Topological embedding: This is to say (the following expressions are equivalent)
- 1) the manifold topology of N coincides with its subspace topology induced from the topological space M,
- 2) the map is a homeomorphism onto its image.
Homeomorphism is an isomorphism of topological spaces. It implies the map is bijective, and both the map and its inverse are continuous. "onto its image" implies the bijection happens between the domain and the image of the function, not the entire codomain (not the entire embedding space or the entire manifold it embeds in).
The equivalence follows in this way: "the map is a homeomorphism onto its image" -> "topological properties are preserved under the map" -> "a set is open in the submanifold if and only if it is open in the larger manifold when viewed as a subset" -> the way we define "nearness" or "continuity" in the submanifold (submanifold topology) is exactly the same as how we define these concepts in the larger manifold (subspace topology) -> "the manifold topology of N coincides with its subspace topology induced from the topological space M"

Proposition: If the set X is a subset of a manifold (M, A+), then it admits at most one submanifold structure.

This can be proved by contradiction: We can assume there are two manifold structures and thus two maximal atlases, and then show that they must be identical. Intuitively, this is because the submanifold's structure is induced by the structure of the embedding space.
However, they are not necessarily identical, since the submanifold generally has fewer dimensions than the embedding space.

Let St(p, n) (p ≤ n) denote the set of all n × p orthonormal matrices; i.e.,
- St(p, n) := {X ∈ R^n×p : X^T X = I_p },
- where I_p denotes the p × p identity matrix.
The set St(p, n), endowed with its submanifold structure to be discussed, is called an (orthogonal or compact) Stiefel manifold. St(p, n) is a (closed) subset of the set R^n×p.
In general, when the embedding space is R^n×p or an open subset of R^n×p, we say that N is a matrix submanifold.
The Stiefel manifold St(p, n) is distinct from the noncompact Stiefel manifold R^n×p_*.
When p = 1, the Stiefel manifold St(p, n) reduces to the unit sphere S^n−1 in R^n. Its dimension is n-1. (The norm of vectors equals to one.)
When p = n, the Stiefel manifold St(p, n) becomes the orthogonal group O_n. Its dimension is n(n−1)/2.

In general, the (orthogonal or compact) Stiefel manifold is the set of all ordered orthonormal p-tuples of vectors in R^n. One common way to organize them is to put them into n × p matrices.
- 1) the order matters, each column of the matrix represents one of the vectors, and the order of the columns reflects the ordering of the vectors.
- 2) orthonormal = orthogonal + unit length. Orthogonality implies linear independence, but the converse is not true.
- 3) the columns form an orthonormal basis for a k-dimensional subspace of R^n.
- 4) compact: thanks to orthonormal. In general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space.
The noncompact Stiefel manifold is defined as the set of all linearly independent p-tuples of vectors in R^n. It is the set of all n × p matrices of full column rank.
- 1) the order is implicitly encoded in the columns of matrices.
- 2) not necessary to be orthogonal or of unit length.
- 3) the columns form a basis for a k-dimensional subspace of R^n.
- 4) noncompact: it is an open subset of n × p matrices.
It is important to remember that the Stiefel manifold is a geometric object -- a space of points -- where each point represents an ordered set of vectors.
(From wikipedia: The compact Stiefel manifold is a deformation retract of the non-compact one, by Gram–Schmidt orthonormalization. They are homotopy equivalent.)

Last updated 1 year ago