Flat (geometry)

In mathematics flat surface spaces are Riemannian manifolds as flax (ger .: flats called). The term is particularly important in the theory of non-positive curvature and especially in the theory of symmetrical spaces .

definition

Let it be a simply connected Riemann manifold. A -dimensional total-geodetic submanifold is an r-flat if it is isometric to Euclidean space . From this it follows in particular that the section curvature of this submanifold is constantly zero. ${\displaystyle X}$ ${\displaystyle r}$ ${\displaystyle F\subset X}$ ${\displaystyle \mathbb {R} ^{r}}$

Flax can also be characterized as simply connected, total-geodetic submanifolds, the curvature of which is constantly zero.

The maximum dimension flax in a manifold is called the maximum flax in . ${\displaystyle X}$ ${\displaystyle X}$

Examples

In Euclidean space the affine subspaces are the only flax. There are other submanifolds with vanishing sectional curvature (e.g. circular cylinder im ), but these are not simply connected and therefore not isometric to a Euclidean space. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{3}}$

In manifolds with negative sectional curvature, the geodesics (1-dimensional flax) are already the maximum flax, since 2-dimensional flax would have vanishing curvature.

Flax in symmetrical spaces

Let it be a symmetrical space of non-compact type of rank , and its Cartan decomposition . For let the exponential map in . ${\displaystyle X=G/K}$ ${\displaystyle r}$ ${\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}$ ${\displaystyle x\in X}$ ${\displaystyle exp_{x}\colon {\mathfrak {p}}\to X}$ ${\displaystyle x}$

Then all containing -flax are of the shape ${\displaystyle x}$ ${\displaystyle r}$ ${\displaystyle F\subset X}$

${\displaystyle F=exp_{x}({\mathfrak {a}})}$

for a maximally Abelian subalgebra . ${\displaystyle {\mathfrak {a}}\subset {\mathfrak {p}}}$

(In particular, the term Weyl chambers can be applied to flax in symmetrical spaces.)

Furthermore, there are two flax and every two points is an isometric view with ${\displaystyle F_{1},F_{2}\subset X}$ ${\displaystyle x_{1}\in F_{1},x_{2}\in F_{2}}$ ${\displaystyle g\in G}$

${\displaystyle g(F_{1})=F_{2},g(x_{1})=x_{2}}$.

The rank of a symmetrical space is (by definition) the dimension of a maximum flax.

See also

• Flat manifold

literature

• Sigurdur Helgason: Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34th American Mathematical Society, Providence, RI, 2001. ISBN 0-8218-2848-7
• Patrick Eberlein: Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1996. ISBN 0-226-18197-9 ; 0-226-18198-7