Glossary of differential geometry and topology

• Bundle – see fiber bundle. • Basic element – A basic element x with respect to an element y is an element of a cochain complex (C^*, d) (e.g., complex of differential forms on a manifold) that is closed: dx = 0 and the contraction of x by y is zero. ==C==

• Characteristic class • Chart • Cobordism • Codimension – The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold. • Connected sum • Connection • Cotangent bundle – the vector bundle of cotangent spaces on a manifold. • Cotangent space • Covering • Cusp • CW-complex ==D==

• Dehn twist • Diffeomorphism – Given two differentiable manifolds M and N, a bijective map f from M to N is called a diffeomorphism – if both f:M\to N and its inverse f^{-1}:N\to M are smooth functions. • Differential form • Domain invariance • Doubling – Given a manifold M with boundary, doubling is taking two copies of M and identifying their boundaries. As the result we get a manifold without boundary. ==E==

• Embedding • Exotic structure – See exotic sphere and exotic \R^4. ==F==

• Fiber – In a fiber bundle, \pi:E \to B the preimage \pi^{-1}(x) of a point x in the base B is called the fiber over x, often denoted E_x. • Fiber bundle • Frame – A frame at a point of a differentiable manifold M is a basis of the tangent space at the point. • Frame bundle – the principal bundle of frames on a smooth manifold. • Flow ==G==

• Genus • Germ • Grassmannian bundle • Grassmannian manifold ==H==

• Handle decomposition • Hypersurface – A hypersurface is a submanifold of codimension one. ==I==

• Immersion • Integration along fibers • Irreducible manifold • Isotopy == J ==

• Jet • Jordan curve theorem ==L==

• Lens space – A lens space is a quotient of the 3-sphere (or (2n + 1)-sphere) by a free isometric action of Z – k. • Local diffeomorphism ==M==

• Manifold – A topological manifold is a locally Euclidean Hausdorff space (usually also required to be second-countable). For a given regularity (e.g. piecewise-linear, C^k or C^\infty differentiable, real or complex analytic, Lipschitz, Hölder, quasi-conformal...), a manifold of that regularity is a topological manifold whose charts transitions have the prescribed regularity. • Manifold with boundary • Manifold with corners • Mapping class group • Morse function ==N==

• Neat submanifold – A submanifold whose boundary equals its intersection with the boundary of the manifold into which it is embedded. == O ==

• Orbifold • Orientation of a vector bundle ==P==

• Pair of pants – An orientable compact surface with 3 boundary components. All compact orientable surfaces can be reconstructed by gluing pairs of pants along their boundary components. • Parallelizable – A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial. • Partition of unity • PL-map • Poincaré lemma • Principal bundle – A principal bundle is a fiber bundle P \to B together with an action on P by a Lie group G that preserves the fibers of P and acts simply transitively on those fibers. • Pullback == R ==

• Section • Seifert fiber space • Submanifold – the image of a smooth embedding of a manifold. • Submersion • Surface – a two-dimensional manifold or submanifold. • Systole – least length of a noncontractible loop. ==T==

• Tangent bundle – the vector bundle of tangent spaces on a differentiable manifold. • Tangent field – a section of the tangent bundle. Also called a vector field. • Tangent space • Thom space • Torus • Transversality – Two submanifolds M and N intersect transversally if at each point of intersection p their tangent spaces T_p(M) and T_p(N) generate the whole tangent space at p of the total manifold. • Triangulation • Trivialization • Tubular neighborhood ==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. ==W==

• Whitney sum – A Whitney sum is an analog of the direct product for vector bundles. Given two vector bundles \alpha and \beta over the same base B their cartesian product is a vector bundle over B\times B. The diagonal map B\to B\times B induces a vector bundle over B called the Whitney sum of these vector bundles and denoted by \alpha \oplus \beta. • Whitney topologies ==References==