Cutting curvature

The cutting curvature is a quantity of Riemannian geometry , a branch of mathematics . With their help one can describe the curvature of a -dimensional Riemannian manifold . Each (two-dimensional) plane in the tangential space is assigned a number as a curvature at a point of this manifold. The section curvature can be understood as a generalization of the Gaussian curvature . The name comes from the fact that a section is made through the manifold in the direction of the given plane and the Gaussian curvature of the resulting surface is determined. ${\displaystyle n}$

definition

A Riemannian manifold , a point in and a two-dimensional subspace (plane) of the tangent space of in the point are given . Let be and two tangent vectors that span this plane. With ${\displaystyle (M,g)}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle \sigma }$ ${\displaystyle T_{p}M}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle v}$ ${\displaystyle w}$

${\displaystyle |v\wedge w|={\sqrt {g(v,v)g(w,w)-g(v,w)^{2}}}}$

denotes the area of the parallelogram spanned by and , denotes the Riemann curvature tensor . ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle R}$

Then the size depends

${\displaystyle K(v,w)={\frac {g(R(v,w)w,v)}{|v\wedge w|^{2}}}={\frac {g(R(v,w)w,v)}{g(v,v)g(w,w)-g(v,w)^{2}}}}$

only on the plane , but not on the choice of vectors and spanning it . Therefore one writes for also and calls this the cutting curvature of . ${\displaystyle \sigma }$ ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle K(v,w)}$ ${\displaystyle K(\sigma )}$ ${\displaystyle \sigma }$

Since there are different sign conventions for the Riemann curvature tensor , the sectional curvature is also determined by

${\displaystyle K(v,w)={\frac {g(R(v,w)v,w)}{|v\wedge w|^{2}}}}$

Are defined. However, this article uses the first convention.

In local coordinates, the above formula for the section curvature can also be written as follows:

${\displaystyle K(v,w)={\frac {R_{\alpha \beta \gamma \delta }v^{\alpha }w^{\beta }v^{\gamma }w^{\delta }}{g_{\mu \nu }v^{\mu }v^{\nu }g_{\mu \nu }w^{\mu }w^{\nu }-(g_{\mu \nu }v^{\mu }w^{\nu })^{2}}}}$

Relationship to Gaussian curvature

Let be a 2-dimensional submanifold of Euclidean space and the on induced metric. For each point and base of is the curvature of intersection ${\displaystyle M\subset \mathbb {R} ^{3}}$ ${\displaystyle g}$ ${\displaystyle M}$ ${\displaystyle p\in M}$ ${\displaystyle \{v,w\}}$ ${\displaystyle \sigma =T_{p}M}$

${\displaystyle K(\sigma )=K(v,w)={\frac {g(R(v,w)w,v)}{g(v,v)g(w,w)-g(v,w)^{2}}}}$

equal to the Gaussian curvature of in the point . The fact that the Gaussian curvature can be represented in this way is a consequence of Gauss' Theorema egregium . ${\displaystyle K(p)}$ ${\displaystyle M}$ ${\displaystyle p}$

Relationships to other curvature quantities

• All information provided by the Riemann curvature tensor is contained in the section curvature. The Riemann curvature tensor can therefore be recovered from the sectional curvature. Namely, be and two - tensors , which are the symmetry properties ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$ ${\displaystyle (0,4)}$

${\displaystyle R_{i}(W,X,Y,Z)=-R_{i}(X,W,Y,Z)}$, , ${\displaystyle R_{i}(W,X,Y,Z)=-R_{i}(W,X,Z,Y)}$ ${\displaystyle R_{i}(W,X,Y,Z)=R_{i}(Y,Z,W,X)}$

and the Bianchi identity

${\displaystyle R_{i}(W,X,Y,Z)+R_{i}(X,Y,W,Z)+R_{i}(Y,W,X,Z)=0}$

fulfill. Then the equation holds for every pair of linearly independent vectors ${\displaystyle X,Y}$

${\displaystyle {\frac {R_{1}(X,Y,Y,X)}{|X|^{2}|Y|^{2}-\langle X,Y\rangle ^{2}}}={\frac {R_{2}(X,Y,Y,X)}{|X|^{2}|Y|^{2}-\langle X,Y\rangle ^{2}}},}$

so follows . ${\displaystyle R_{1}=R_{2}}$

• Since one can recover the Riemann curvature tensor from the section curvature , one can also find a relationship between the Ricci curvature and the section curvature. Let an orthonormal basis of the tangent space hold ${\displaystyle R}$ ${\displaystyle K}$ ${\displaystyle \operatorname {Ric} }$ ${\displaystyle (E_{1},\ldots ,E_{n})}$ ${\displaystyle T_{p}M}$

${\displaystyle \operatorname {Ric} (E_{1},E_{1})=\sum _{k=1}^{n}g(R(E_{k},E_{1})E_{1},E_{k})=\sum _{k=2}^{n}K(E_{1},E_{k}).}$

The Ricci curvature is completely determined by the formula, since the Ricci tensor is symmetric . If the underlying, Riemannian manifold of the dimension has constant sectional curvature, the simplified formula applies ${\displaystyle (M,g)}$ ${\displaystyle n}$

${\displaystyle \operatorname {Ric} (V,V)=(n-1)\,K\,g(V,V).}$

• A similar formula is obtained for the scalar curvature ${\displaystyle S}$

${\displaystyle S=\sum _{j,k=1}^{n}g(R(E_{k},E_{j})E_{j},E_{k})=\sum _{j\neq k}^{n}K(E_{j},E_{k}),}$

where is again an orthonormal basis of the tangent space. If the cutting curvature is constant, then the following applies ${\displaystyle (E_{1},\ldots ,E_{n})}$

${\displaystyle S=n(n-1)\,K.}$

Examples

• The sectional curvature of Euclidean space is constant zero, because the Riemannian curvature tensor is defined in such a way that it vanishes for all points from . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$
• The sphere with radius has curvature of intersection . Since this is isotropic and homogeneous, the cutting curvature is constant and it is sufficient to determine this at the North Pole . With that is exponential in the North Pole called. In addition, let the two-dimensional subspace of the tangential space , which is spanned by. Now is a manifold which is isometric to . It is known from this that the Gaussian curvature is. Hence the -dimensional sphere also has the curvature of intersection . ${\displaystyle \mathbb {S} _{R}^{n}}$ ${\displaystyle R}$ ${\displaystyle {\tfrac {1}{R^{2}}}}$ ${\displaystyle N}$ ${\displaystyle \exp _{N}}$ ${\displaystyle \Pi \subset T_{N}\mathbb {S} _{R}^{n}}$ ${\displaystyle T_{N}\mathbb {S} _{R}^{n}}$ ${\displaystyle \{\partial _{1},\partial _{2}\}}$ ${\displaystyle \exp _{N}(\Pi )}$ ${\displaystyle \mathbb {S} _{R}^{2}}$ ${\displaystyle \textstyle {\frac {1}{R^{2}}}}$ ${\displaystyle n}$ ${\displaystyle \textstyle {\frac {1}{R^{2}}}}$
• The hyperbolic space has section curvature ${\displaystyle \mathbb {H} _{R}^{n}}$ ${\displaystyle \textstyle -{\frac {1}{R^{2}}}.}$

Applications

Manifolds with constant curvature (from left to right): the hyperboloid with negative curvature, the cylinder with zero curvature, and the sphere with positive curvature.

Manifolds with constant curvature

As in other sub-areas of mathematics, one tries to classify objects in Riemannian geometry. The corresponding Riemannian manifolds are classified in Riemannian geometry. So two manifolds are understood to be the same if there is an isometric map between them. Because it depends on the Riemannian metric, the section curvature is an important invariant of Riemannian manifolds. In the case of complete , simply connected Riemannian manifolds with constant section curvature, the classification is relatively simple, because there are only three cases to be considered. If the Riemannian manifold has the dimension and the constant, positive section curvature , then it is isometric (equal) to the -dimensional sphere with radius . If the section curvature is constantly zero, then the manifold is called flat and it is isometric to Euclidean space and in the case that the manifold has the negative section curvature , it corresponds to the -dimensional hyperbolic space . ${\displaystyle n}$ ${\displaystyle \textstyle {\frac {1}{R^{2}}}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {S} _{R}^{n}}$ ${\displaystyle R}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \textstyle -{\frac {1}{R^{2}}}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {H} _{R}^{n}}$

If one no longer only considers the simply connected manifolds, but all complete and connected manifolds with constant intersection curvature, their classification is already more complicated. The fundamental group of these manifolds no longer disappears. It can now be shown that such manifolds are isometric too . Where for one of the three spaces from the above section stands for or and is a discrete subgroup of the isometric group of , which operates freely and actually discontinuously on . This group is isomorphic to the fundamental group of . ${\displaystyle M}$ ${\displaystyle N/\Gamma }$ ${\displaystyle N}$ ${\displaystyle \mathbb {S} _{R}^{n},\,\mathbb {R} ^{n}}$ ${\displaystyle \mathbb {H} _{R}^{n}}$ ${\displaystyle \Gamma }$ ${\displaystyle J(N)}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle \Gamma }$ ${\displaystyle \pi _{1}(M)}$ ${\displaystyle M}$

Manifolds with negative curvature

In 1928, Élie Cartan generalized a result by Jacques Hadamard , which states in a modern formulation that the exponential mapping is a universal superposition in the case of non-positive sectional curvature . This statement is now called the Cartan-Hadamard Theorem . There are different formulations of the sentence. The version for Riemannian manifolds is precisely:

Is a complete , connected Riemannian manifold, all of whose intersection curvatures are not positive. Then the exponential map is a universal overlay map for all . In particular, the overlay space is diffeomorphic to . If it is even simply connected , then it is diffeomorphic to . ${\displaystyle M}$ ${\displaystyle \operatorname {exp} _{p}:T_{p}M\to M}$ ${\displaystyle p\in M}$ ${\displaystyle T_{p}M}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$

This theorem is remarkable, among other things, because it provides a connection between a local quantity and a global quantity of a differentiable manifold. Such statements are also called local-global theorems . In this case, the curvature of intersection of the manifold is the local quantity, because the curvature of intersection is defined for each . Assuming that the manifold is simply connected, according to the theorem it is diffeomorphic to , which is a global, differential topological property that has nothing to do with the Riemannian metric. From the theorem it follows that compact, complete, simply connected manifolds, such as the sphere is one, always have to have a positive intersection curvature somewhere. Because, because the sphere is compact, it cannot be diffeomorphic to be. The condition of the non-positive section curvature therefore gives rise to strong restrictions with regard to the topology that the manifold can support. With the aid of algebraic topology it can be shown that the homotopy groups of the manifolds that meet the requirements of the theorem vanish for. ${\displaystyle p\in M}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \pi _{k}(M)}$ ${\displaystyle k>1}$

Manifolds with positive curvature

One result from the area of ​​manifolds with positive section curvature is Bonnet's theorem . This local-global theorem connects the section curvature with the topological properties compactness and finite fundamental group. The sentence says precisely:

Let be a complete, connected Riemannian manifold. All section curvatures are limited by downwards. Then there is a compact space with a finite fundamental group . ${\displaystyle M}$ ${\displaystyle \textstyle {\frac {1}{R^{2}}}}$ ${\displaystyle M}$

literature

• John M. Lee: Riemannian Manifolds. An Introduction to Curvature. Springer, New York 1997, ISBN 0387983228 , chapter 8.
• Manfredo Perdigão do Carmo: Riemannian Geometry , Birkhäuser, Boston 1992, ISBN 0-8176-3490-8 , chapter 4.3.

Individual evidence

1. ↑ John M. Lee: Riemannian Manifolds. An Introduction to Curvature. Springer, New York 1997, ISBN 0387983228 , page 146.
2. ↑ Manfredo Perdigão do Carmo: Riemannian Geometry , Birkhäuser, Boston 1992, ISBN 0-8176-3490-8 , page 94.