8 H

Hypersurface. A hypersurface is a submanifold of codimension one.

9 I

ImmersionIn mathematics, an embedding is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. Topology/Geometry General topology In general topology, an embedding is a homeomorphism onto its image. More ex

10 L

Lens spaceA lens space is an example of a topological space, considered in mathematics. Definition Sit the sphere inside as the set of all -tuples of unit absolute value. Let be a primitive th root of unity and let : be integers relatively prime to. Let the set of. A lens space is a quotient of the 3-sphere (or (2n+1)-sphere) by a free isometric action of Z_k.

11 M

Manifold. A topological manifold is a locally Eulidean Hausdorff space. (In Wikipedia, a manifold need not be paracompact or second-countable.) A C^k manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A C^∞ or smooth manifold is a differentiable manifold whose chart overlap functions are infinitely continuously differentiable.

12 P

Parallelizable. A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial.

Principal bundle. A principal bundle is a fiber bundle P → B together with right action on P by a Lie group G that preverses the fibers of P and acts simply transitively on those fibers.

Pullback

13 S

Section

Submanifold. A submanifold is the image of a smooth embedding of a manifold.

Submersion

Surface, a two-dimensional manifold or submanifold.

14 T

Tangent bundle, the vector bundle of tangent spaces on a differtiable manifold.

Tangent field, a section of the tangent bundle. Also called a vector field.

Tangent space

Torus

Transversality. Two submanifolds M and N intersect transversally if at each point of intersection p their tangent spaces and generate the whole tangent space at p of the total manifold.

Trivialization

15 V

Vector bundle, a fiber bundle whose fibers are vector spaces and whose transition functions are linear maps.

Vector field, a section of a vector bundle. More specifically, a vector field can mean a section of the tangent bundle.

16 W

Whitney sum. A Whitney sum is an analog of the direct product for vector bundles. Given two vector bundles α and β over the same base B their cartesian product is a vector bundle over B ×B. The diagonal map induces a vector bundle over B called the Whitney sum of these vector bundles and denoted by α⊕β.

Differential geometry Differential topology Differential geometry

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