Two

theorems In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

in the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

bear the name Myers–Steenrod theorem, both from a 1939 paper by

Myers Myers as a surname has several possible origins, e.g. Old French ("physician"), Old English ("mayor"), and Old Norse ("marsh"). People * Abram F. Myers (1889–after 1960), chair of the Federal Trade Commission and later general counsel and b ...

and Steenrod. The first states that every distance-preserving surjective map (that is, an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

s) between two

connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...

Riemannian manifolds In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the n-sphere, hyperbolic space, and smooth surfaces in ...

is a smooth

of Riemannian manifolds. A simpler proof was subsequently given by

Richard Palais Richard Sheldon Palais (born May 22, 1931) is an American mathematician working in differential geometry. Education and career Palais studied at Harvard University, where he obtained a B.A. in 1952, an M.A. in 1954 and a Ph.D. in 1956. His Ph ...

in 1957. The main difficulty lies in showing that a distance-preserving map, which is a priori only

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

, is actually

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

. The second theorem, which is harder to prove, states that the

isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element ...

\mathrm(M)

of a connected

\mathcal^2

Riemannian manifold

M

is a

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

in a way that is compatible with the

compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory ...

and such that the action

\mathrm(M)\times M \longrightarrow M

\mathcal^1

differentiable (in both variables). This is a generalization of the easier, similar statement when

M

is a

Riemannian symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geomet ...

: for instance, the group of isometries of the two-dimensional unit

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

is the

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

O(3)

. A harder generalization is given by the Bochner-Montgomery theorem, where

\mathrm(M)

is replaced by a locally compact transformation group of diffeomorphisms of

M

References

* * Theorems in Riemannian geometry {{Riemannian-geometry-stub