Bonnet theorem
Encyclopedia
In the mathematical field of differential geometry, more precisely, the theory of surface
s in Euclidean space
, the Bonnet theorem states that the first
and second fundamental forms determine a surface in R3 uniquely up to a rigid motion. It was proven by Pierre Ossian Bonnet
in about 1860.
This is not to be confused with the Bonnet–Myers theorem or Gauss–Bonnet theorem
.
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, the Bonnet theorem states that the first
First fundamental form
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3. It permits the calculation of curvature and metric properties of a surface such as length and...
and second fundamental forms determine a surface in R3 uniquely up to a rigid motion. It was proven by Pierre Ossian Bonnet
Pierre Ossian Bonnet
Pierre Ossian Bonnet was a French mathematician. He made some important contributions to the differential geometry of surfaces, including the Gauss-Bonnet theorem.-Early years:...
in about 1860.
This is not to be confused with the Bonnet–Myers theorem or Gauss–Bonnet theorem
Gauss–Bonnet theorem
The Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
.