Blocking light in compact Riemannian manifolds

Lafont, Jean-François; Schmidt, Benjamin

doi:10.2140/gt.2007.11.867

Cited by 15 publications

(32 citation statements)

References 18 publications

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“…Berger [Be78] proved that a Blaschke manifold diffeomorphic to the sphere is in fact isometric to a round sphere. This was used in [LS07] to prove Theorem 1 for Blaschke manifolds.…”

Section: Theorem 1 a Closed Riemannian Manifold M Has Cross And Sphementioning

confidence: 99%

“…In [LS07] it was also conjectured that a closed Riemannian manifold with cross blocking is isometric to a compact rank one symmetric space. We prove that this is the case in dimension two: Section 1 contains some preliminary material concerning Morse theory for path spaces and properties of totally convex subsets in Riemannian manifolds.…”

Section: Theorem 1 a Closed Riemannian Manifold M Has Cross And Sphementioning

confidence: 99%

“…More recently, blocking light has been studied in Riemannian spaces (see e.g. [BaGu], [BG], [GB], [GS06], [He], and [LS07]). Here we give a characterization of the round sphere in terms of its blocking properties.…”

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A characterization of round spheres in terms of blocking light

Schmidt¹,

Souto²

2010

Comment. Math. Helv.

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Abstract.A closed Riemannian manifold M is said to have cross (compact rank one symmetric space) blocking if whenever p ¤ q are less than the diameter apart, all light rays from p can be shaded away from q with at most two point shades. Similarly, a closed Riemannian manifold is said to have sphere blocking if for each p 2 M all the light rays from p are shaded away from p by a single point shade. We prove that Riemannian manifolds with cross and sphere blocking are isometric to round spheres. Mathematics Subject Classification (2000). Riemannian manifold, cross blocking, sphere blocking, blocking number.Keywords. 53C20, 53C22, 37D40.In this note we characterize constant curvature spheres in terms of light blocking properties.Definition (Light). Let X; Y be two nonempty subsets of a Riemannian manifold M , and let G M .X; Y / denote the set of unit speed parametrized geodesics W OE0; L ! M with initial point .0/ 2 X and terminal point .L / 2 Y . The light from X to Y is the setIntuitively, we are postulating that X emits light traveling along geodesics, that Y consists of receptors, and that X and Y are opaque while the remaining medium M n fX [ Y g is transparent. From this point of view, L M .X; Y / is the set of light rays from X to Y and a set Z blocks the light from X to Y if it completely shades X away from Y . This simple model ignores diffraction, the dual nature of light, and all aspects of quantum mechanics.A well-known result of Serre [Se51] asserts that for closed M and points x; y 2 M , the set G M .x; y/ of geodesic segments joining x and y is always infinite. In contrast, 260 B. Schmidt and J. Souto CMH L M .x; y/ is sometimes infinite and sometimes not. For instance, if x and y are different points on the standard round sphere S n with distance less than , then L S n .x; y/ consists of exactly two elements. In particular, we see that, under the same assumptions, it suffices to declare two additional points in S n to be opaque in order to block all the light rays from x to y. Definition (Blocking number). Let x; y 2 M be two (not necessarily distinct) points in M . The blocking number b M .x; y/ for L M .x; y/ is defined byThe study of blocking light (also known as security) seems to have originated in the study of polygonal billiard systems and translational surfaces (see e.g. If x, y are two distinct points in the standard round sphere S n closer than then, as remarked above, b S n .x; y/ Ä 2. This property does not characterize the round sphere amongst all closed Riemannian manifolds. In fact, every compact rank one symmetric space, or CROSS for short, has the following property:Cross blocking: For every pair of distinct points x; y 2 M with d M .x; y/ < diam.M /, we have b M .x; y/ Ä 2.Apart from cross blocking, the round sphere also has the following property:Sphere blocking: For every point x 2 M , we have b M .x; x/ D 1.The CROSSes are classified and consist of the round spheres S n , the projective spaces KP n where K denotes one of R, C, or H, and the Cayley projective plane, each one end...

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“…Berger [Be78] proved that a Blaschke manifold diffeomorphic to the sphere is in fact isometric to a round sphere. This was used in [LS07] to prove Theorem 1 for Blaschke manifolds.…”

Section: Theorem 1 a Closed Riemannian Manifold M Has Cross And Sphementioning

confidence: 99%

Section: Theorem 1 a Closed Riemannian Manifold M Has Cross And Sphementioning

confidence: 99%

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A characterization of round spheres in terms of blocking light

Schmidt¹,

Souto²

2010

Comment. Math. Helv.

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“…See [BG08], [LS07], and [Gut09]. This approach works well under additional assumptions, e. g., that the manifold (M, g) has no conjugate points.…”

Section: Theorem 1 a Two-dimensional Riemannian Torus Is Secure If Amentioning

confidence: 99%

“…The only examples so far are the flat manifolds [GS06]. Researchers in the subject believe in the following statement [BG08,LS07].…”

Section: Introductionmentioning

confidence: 99%

Insecurity for compact surfaces of positive genus

Bangert

Gutkin

2009

Geom Dedicata

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A pair of points in a riemannian manifold $M$ is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in $M$ are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.Comment: 37 pages, 11 figure

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Topological entropy and blocking cost for geodesics in Riemannian manifolds

Gutkin

2008

Geom Dedicata

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Abstract. For a pair of points x, y in a compact, riemannian manifold M let n t (x, y) (resp. s t (x, y)) be the number of geodesic segments with length ≤ t joining these points (resp. the minimal number of point obstacles needed to block them). We study relationships between the growth rates of n t (x, y) and s t (x, y) as t → ∞. We derive lower bounds on s t (x, y) in terms of the topological entropy h(M ) and its fundamental group. This strengthens the results of Burns-Gutkin [2] and Lafont-Schmidt [13]. For instance, by [2,13], h(M ) > 0 implies that s is unbounded; we show that s grows exponentially, with the rate at least h(M )/2.

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Blocking light in compact Riemannian manifolds

Cited by 15 publications

References 18 publications

A characterization of round spheres in terms of blocking light

A characterization of round spheres in terms of blocking light

Insecurity for compact surfaces of positive genus

Topological entropy and blocking cost for geodesics in Riemannian manifolds

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