Dictionary of Meaning
<<Back
Please select a letter:
A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
K |
L |
M |
N |
O |
P |
Q |
R |
S |
T |
U |
V |
W |
X |
Y |
Z |
0-9
Click here for Shopping
Gauss Map
*** Shopping-Tip: Gauss Map
In
differential geometry, the '''Gauss map''' (named, like
List of topics named after Carl Friedrich Gauss so many things, after
Carl F. Gauss) maps a
surface in
Euclidean space '''R'''
3 to the unit
sphere ''S''
2. Namely, given a surface ''X'' lying in '''R'''
3, the Gauss map is a continuous map ''N'': ''X'' → ''S''
2 such that ''N''(''p'') is orthogonal to ''X'' at ''p''.
The Gauss map can be defined (globally) if and only if the surface is
orientable, but it is always defined locally (i.e. on a small piece of the surface). The
Jacobian of the Gauss map is equal to
Curvature Gauss curvature, and the
differential (mathematics) differential of the Gauss map is called the
shape operator.
Simplified explanation
Each point on the surface has a
normal vector; that is, a vector
orthogonal to the surface at that point. Now, move this vector to the origin. Do this for all such vectors on the surface. What we get is a surface on the 2-sphere (possibly with overlaps). This is called the '''Gauss map'''. A similar concept in 2
dimensions with curves is the
radial of a curve.
Generalizations
The '''Gauss map''' can be defined the same way for
hypersurfaces in '''R'''
''n'', this way we get a map from a hypersurface to the unit sphere ''S''
''n'' − 1 ∈ '''R'''
''n''.
For a general oriented ''k''-
submanifold of '''R'''
''n'' the Gauss map can be also be defined, and its target space is the ''oriented''
Grassmannian
, i.e. the set of all oriented ''k''-planes in '''R'''
''n''. In this case a point on the submanifold is mapped to its oriented tangent subspace. It should be noted that in
Euclidean space Euclidean 3-space, an oriented 2-plane is characterized by a unit normal vector, hence this is consistent with the definition above.
Finally, the notion of Gauss map can be generalized to an oriented submanifold ''X'' of dimension ''k'' in an oriented ambient
Riemannian manifold ''M'' of dimension ''n''. In that case, the Gauss map then goes from ''X'' to the set of tangent ''k''-planes in the
tangent bundle ''TM''. The target space for the Gauss map ''N'' is a
Grassmann bundle built on the tangent bundle ''TM''.
Category:Differential geometry
Category:Riemannian geometry
see
Gauss map
