Immersion (mathematics) - Examples and Properties

Examples and Properties

• The Klein bottle, and all other non-orientable closed surfaces, can be immersed in 3-space but not embedded.

• A mathematical rose with k petals is an immersion of the circle in the plane with a single k-tuple point; k can be any odd number, but if even must be a multiple of 4, so the figure 8 is not a rose.
• By the Whitney–Graustein theorem the regular homotopy classes of immersions of the circle in the plane are classified by the winding number which is also the number of double points counted algebraically (i.e. with signs).
• The sphere can be turned inside out: the standard embedding f : S2 → R3 is related to f₁ = −f₀ : S2 → R3 by a regular homotopy of immersions f_t : S2 → R3.
• Boy's surface is an immersion of the real projective plane in 3-space; thus also a 2-to-1 immersion of the sphere.
• The Morin surface is an immersion of the sphere; both it and Boy's surface arise as midway models in sphere eversion.

• Boy's surface

• The Morin surface