S2×R

S2×R
Name	S2×R
Type	product manifold
Curvature	variable

S2×R

S2×R is the Cartesian product of the 2-sphere S^2 and the real line R, forming a three-dimensional product manifold. It appears in contexts ranging from classical differential topology to geometric structures studied by William Thurston, René Thom, Hermann Weyl, and in examples used by researchers at institutions such as Princeton University, University of Cambridge, ETH Zurich, and Massachusetts Institute of Technology. The manifold is a basic example in texts by John Milnor, Michael Atiyah, Mikhail Gromov, and Peter Scott.

Definition and Notation

S2×R is defined as the set of ordered pairs (p,t) with p in S^2 and t in R equipped with the product topology and smooth structure induced from S^2 and R. Standard notation uses symbols from the literature of Lefschetz, Hatcher, Milnor, and Spanier; coordinates often derive from charts used by Christoffel and constructions in Riemannian geometry found in works by Élie Cartan and Bernhard Riemann. The projection maps π1: S2×R → S^2 and π2: S2×R → R are smooth surjections analogous to maps in fibrations treated by Serre and Eilenberg–MacLane.

Topology and Geometry

Topologically, S2×R is noncompact and orientable; it has the same local structure as R^3 but global structure like a product treated in examples by Leray and de Rham. The manifold admits embeddings and immersions related to results of Whitney and isotopy theorems discussed by Smale and Hirsch. Important comparisons are often made with the three-sphere S^3, the three-dimensional Euclidean space R^3, and product spaces such as S^1×S^2 and H^2×R considered by Thurston and Scott. In the context of compactifications used in Alexander duality and constructions in Poincaré conjecture literature, S2×R provides canonical counterpoints to closed manifolds studied by Perelman.

Fundamental Group and Homology

The fundamental group π1(S2×R) is isomorphic to π1(S^2)×π1(R), hence trivial, reflecting classical results of Van Kampen and computations found in Hatcher and Spanier. Homology groups H_n(S2×R;Z) match those of S^2 up to degree shifts as explained in texts by Bott and Tu: H_2 ≅ Z, H_1 ≅ 0, and H_0 ≅ Z, aligning with calculations in the context of Alexander duality and examples in Eilenberg–Steenrod axioms. Cohomology rings and cup product structure follow from the Künneth theorem proven by Künneth and elaborated by Mac Lane and Cartan.

Covering Spaces and Fibrations

S2×R is itself a trivial bundle S^2×R with projection maps serving as fibrations in the sense of Serre and Hurewicz. Its covering spaces include universal covers studied in works by Hatcher and examples in Hempel; the universal cover is homeomorphic to S2×R because π1 is trivial, paralleling situations analyzed by Deck and Hopf. Nontrivial bundles such as the nontrivial S^2-bundle over S^1 (often compared in literature by Milnor and Stiefel–Whitney theory) contrast with S2×R, which admits product trivializations used in construction methods of Thom and Pontryagin.

Possible Metrics and Curvature

S2×R admits product Riemannian metrics combining a round metric on S^2 (as in treatments by Gauss and Hopf) with the Euclidean metric on R; these metrics are discussed in monographs by Chavel and Petersen. Sectional curvature can be positive in S^2 directions and zero along the R-direction, a situation appearing in classifications by Thurston and curvature comparison theorems of Toponogov and Cheeger–Gromoll. Warped product metrics producing metrics with variable curvature are constructed following methods of Besse and examples in the study of Einstein metrics by Yau and Aubin.

Applications and Examples

S2×R appears as a model geometry in the list of geometries studied by Thurston and as ambient space for hypersurfaces studied by Chern and Schoen. It serves as the spatial slice in cosmological models referenced in literature by Einstein and Friedmann when considering product topologies, and as a setting for minimal surface examples in papers by Osserman and Allard. Examples of flows, including Ricci flow applications discussed by Hamilton and Perelman, use S2×R as starting manifolds for surgery and degeneration analyses by Kleiner and Lott.

Related Manifolds and Constructions

Related spaces include S^3, S^1×S^2, R^3, H^2×R, and nontrivial sphere bundles studied by Milnor and Stiefel. Constructions connecting S2×R to lens spaces such as Lens space examples, mapping tori analyzed by Thurston and Fried, and products in the spirit of Künneth and Cup product computations appear throughout the literature of Algebraic topology authors including Spanier, Hatcher, and Brown.

Category:3-manifolds