In
sub-Riemannian geometry
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal s ...
, the Chow–Rashevskii theorem (also known as Chow's theorem) asserts that any two points of a connected sub-Riemannian manifold, endowed with a bracket generating distribution, are connected by a horizontal path in the manifold. It is named after
Wei-Liang Chow
Chow Wei-Liang (; October 1, 1911, Shanghai – August 10, 1995, Baltimore) was a Chinese mathematician and stamp collector born in Shanghai, known for his work in algebraic geometry.
Biography
Chow was a student in the US, graduating from the ...
who proved it in
1939
This year also marks the start of the Second World War, the largest and deadliest conflict in human history.
Events
Below, the events of World War II have the "WWII" prefix.
January
* January 1
** Third Reich
*** Jews are forbidden to ...
, and
Petr Konstanovich Rashevskii, who proved it independently in
1938
Events
January
* January 1
** The Constitution of Estonia#Third Constitution (de facto 1938–1940, de jure 1938–1992), new constitution of Estonia enters into force, which many consider to be the ending of the Era of Silence and the a ...
.
The theorem has a number of equivalent statements, one of which is that the
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
induced by the
Carnot–Carathéodory metric
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal ...
is equivalent to the intrinsic (locally Euclidean) topology of the manifold. A stronger statement that implies the theorem is the
ball–box theorem. See, for instance, and .
See also
*
Orbit (control theory) The notion of orbit of a control system used in mathematical control theory is a particular case of the notion of orbit in group theory.
Definition
Let
\dot q=f(q,u)
be a \ ^\infty control system, where
belongs to a finite-dimensional manifol ...
References
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Metric geometry
Theorems in geometry
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