Beltrami's theorem
Encyclopedia
In mathematics
— specifically, in Riemannian geometry
— Beltrami's theorem is a result named after the Italian
mathematician
Eugenio Beltrami
which states that geodesic map
s preserve the property of having constant curvature
. More precisely, if (M, g) and (N, h) are two Riemannian manifolds and φ : M → N is a geodesic map between them, and if either of the manifolds (M, g) or (N, h) has constant curvature, then so does the other one.
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...
— specifically, in Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
— Beltrami's theorem is a result named after the Italian
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Eugenio Beltrami
Eugenio Beltrami
Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics...
which states that geodesic map
Geodesic map
In mathematics—specifically, in differential geometry—a geodesic map is a function that "preserves geodesics"...
s preserve the property of having constant curvature
Constant curvature
In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points...
. More precisely, if (M, g) and (N, h) are two Riemannian manifolds and φ : M → N is a geodesic map between them, and if either of the manifolds (M, g) or (N, h) has constant curvature, then so does the other one.