In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifold

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

s. It is named after Heinz Hopf

Heinz Hopf

Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...

 and his student Willi Rinow (1907–1979).

Statement of the theorem

Let (M, g) be a connected Riemannian manifold. Then the following statements are equivalent:

1. The closed
   Closed
   Closed may refer to:Math* Closure * Closed manifold* Closed orbits* Closed set* Closed differential form* Closed map, a function that is closed.Other* Cloister, a closed walkway* Closed-circuit television...
   
    and bounded
   Bounded set
   In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
   
    subset
   Subset
   In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
   
   s of M are compact
   Compact space
   In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
   
   ;
2. M is a complete
   Complete space
   In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
   
    metric space
   Metric space
   In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
   
   ;
3. M is geodesically complete; that is, for every p in M, the exponential map
   Exponential map
   In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
   
    exp_p is defined on the entire tangent space
   Tangent space
   In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
   
    T_pM.

Furthermore, any one of the above implies that given any two points p and q in M, there exists a length minimizing geodesic

Geodesic

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

 connecting these two points (geodesics are in general extrema

Extrema

Extrema may refer to:*maxima and minima*Extremities*Extremophiles*Extrema, Minas Gerais, town in Brazil*Extrema, Rondônia, town in Brazil*Extrema , Italian metal band...

, and may or may not be minima).

Variations and generalizations

• The Hopf–Rinow theorem is generalized to length-metric spaces the following way:

  If a length-metric space (M, d) is complete

  Complete space

  In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

  
   and locally compact then any two points in M can be connected by minimizing geodesic

  Geodesic

  In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

  
  , and any bounded closed set in M is compact

  Compact space

  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

  
  .

• The theorem does not hold in infinite dimensions: showed that two points in an infinite dimensional complete Hilbert manifold need not be connected by a geodesic.