Injectivity radius

The injectivity radius is a term from the mathematical branch of differential geometry , more precisely an invariant of Riemannian manifolds . Its importance lies in the fact that points whose distance is less than the injectivity radius can always be connected by an unambiguous shortest connection .

definition

Let be a Riemannian manifold. With the will slice location of designated. Then the injectivity radius at the point is through ${\displaystyle M}$ ${\displaystyle C(p)}$ ${\displaystyle p\in M}$ ${\displaystyle p\in M}$

${\displaystyle i(p):=d(p,C(p))}$

defined wherein the Riemannian distance on designated. ${\displaystyle d}$ ${\displaystyle M}$

The injectivity radius of the Riemann manifold is then through ${\displaystyle i(M)}$ ${\displaystyle M}$

${\displaystyle i(M):=\inf _{p\in M}i(p)=\inf _{p\in M}d(p,C(p)).}$

Are defined. The name injectivity radius is explained by the fact that the exponential mapping is injective on the geodetic ball if and only if applies. ${\displaystyle \exp _{p}}$ ${\displaystyle B_{r}(p)}$ ${\displaystyle r\leq i(p)}$

Equivalently, one can also define the injectivity radius at the point as the largest for that ${\displaystyle p}$ ${\displaystyle \epsilon }$

${\displaystyle exp_{p}\colon B(0,\epsilon )\to B(p,\epsilon )}$

is a diffeomorphism . (Suffice it to say that on is defined and injective .) ${\displaystyle exp_{p}}$ ${\displaystyle B(0,\epsilon )}$

Examples

• Be the unitary sphere . The injectivity radius at each point is , because the exponential mapping maps the open circular disk diffeomorphically from the radius to the complement of the antipodal point . ${\displaystyle S^{2}\subset \mathbb {R} ^{3}}$ ${\displaystyle p\in S^{2}}$ ${\displaystyle \pi }$ ${\displaystyle exp_{p}\colon T_{p}S^{2}\to S^{2}}$ ${\displaystyle \pi }$ ${\displaystyle p}$
• Let be the flat torus obtained from a unit square by identifying opposite sides. Then the injectivity radius is the same at every point , because the exponential mapping is injective on the interior of a square with a center point and edge length . ${\displaystyle T^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ ${\displaystyle p\in T^{2}}$ ${\displaystyle \textstyle {\frac {1}{2}}}$ ${\displaystyle 0\in T_{p}T^{2}}$ ${\displaystyle 1}$

Estimates of the radius of injectivity

• Estimation with -pinned positive curvature (Klingenberg-Sakai): Let be a complete, simply connected Riemannian manifold with intersection curvature , then is . ${\displaystyle \textstyle {\frac {1}{4}}}$ ${\displaystyle (M,g)}$ ${\displaystyle {\tfrac {1}{4}}<K\leq 1}$ ${\displaystyle i(M)\geq \pi }$
• Improvements in the even-dimensional case: If there is a straight-dimensional, complete Riemannian manifold with sectional curvature , then and if it is additionally orientable , even . ${\displaystyle (M,g)}$ ${\displaystyle K\leq 1}$ ${\displaystyle \textstyle i(M)\geq {\frac {\pi }{2}}}$ ${\displaystyle M}$ ${\displaystyle i(M)\geq \pi }$

Constrained geometry

Riemannian manifolds with a positive radius of injectivity and a restricted curvature of section are called manifolds of restricted geometry .

On complete Riemannian manifolds , the injectivity radius at the point continuously depends on . In particular, all compact Riemannian manifolds have bounded geometry. ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle p}$

literature

• Wilhelm Klingenberg : Riemannian geometry. Second edition. de Gruyter Studies in Mathematics, 1. Walter de Gruyter & Co., Berlin, 1995. ISBN 3-11-014593-6
• Jürgen Jost : Riemannian geometry and geometric analysis. Sixth edition. University text. Springer, Heidelberg, 2011. ISBN 978-3-642-21297-0

Individual evidence

1. ↑ Peter Petersen: Riemannian geometry. Second edition. Graduate Texts in Mathematics, 171. Springer, New York, 2006. ISBN 978-0387-29246-5 (Section 9.2)
2. ↑ Klingenberg, W .; Sakai, T .: Injectivity radius estimate for 1/4-pinched manifolds. Arch. Math. (Basel) 34 (1980), no. 4, 371-376.