Topological Folding of Locally Flat Banach Spaces

Transcription

1 It. Joural of Math. Aalysis, Vol. 6, 0, o. 4, Topological Foldig of Locally Flat aach Spaces E. M. El-Kholy *, El-Said R. Lashi ** ad Salama N. aoud ** *epartmet of Mathematics, Faculty of Sciece, Tata Uiversity, Tata, Egypt **epartmet of Mathematics, Faculty of Sciece, El-Miufiya Uiversity, Shebee El-Kom, Egypt Abstract I this paper we expaded the defiitio of the topological foldig to locally flat aach spaces. The we explored the set of sigularities of this type of foldig of -aach maifolds. Also we proved that the composite of ay topological foldigs of locally flat aach spaces eed ot be a topological foldig while the cartezia product of two such foldigs of locally flat aach spaces is agai a topological foldig. These geometrical structure are importat i formulatio the solutios of the problems of Morse theory, variatioal calculus ad hydrodyamic. Keywords: aach spaces, Topological foldig ad Maifolds () Itroductio Local isometries betwee Reimaia maifolds may be characterized as maps that sed geodesic segmets to geodesic segmets of the same legth. Isometric foldigs are likewise characterized by such a property with the differece that we use piecewise geodesic segmets istead of geodesic segmets. Theory of isometric foldigs is itroduced by Robertso [5] who studied the stratificatio determied by the folds or the sigularities. The the theory of isometric foldigs has bee pushed by both S. A. Robertso ad El-Kholy [6]. Also the idea of topological foldig is modeled by both of them o that of isometric foldig but i the absece of metrical structure [7]. The otio of cellular foldig of a cell complex is iveted by El-Kholy ad Al-Khurasai []. After several attempts of geeralizig the otio of isometric which is differetial geometry to the geometric topological oes, regular foldig were studied by Farra, El-Kholy ad Robertso [].

2 008 E. M. El-Kholy, El-Said R. Lashi ad Salama N. aoud () efiitio We first defie the followig subsets of Euclidea -space E for ay > 0. { x E : }, S = { y E : y = } = x ad S are called the uit disc ad the uit sphere i Euclidea -space respectively. Thus S =. It follows from the defiitio that for each x with x 0, there is a uique real umber t ad a uique poit such that x = t y, 0 t. Of course for all y S, 0 = 0 y, see Fig. ().. y S y f ( y ) x = ty 0 S f Fig. () t f ( y) 0 S Now suppose that f* :, give by: f : S S is ay map. The f iduces a map f* ( t x ) = t f ( x ), 0 t, x ad f * (0) = 0. y usig this costructio we ca defie a topological foldig of locally flat aach spaces by the followig iductio: Let M ad N be a locally flat aach spaces. For all homeomorphism ξ : V a ope set of x x M, a disc chart at x is a, where V x, is a bh. of x i M (homeomorphic to E ) ad ξ (0) = x. Hece every x M has a disc chart. Now let φ : M N be a cotiuous map. We say that φ is a topological foldig of M ito N iff for each x M, there are disc charts ξ : Vx for M at x ad η : W for N at y = φ ( x ) together with a topological foldig y f : S S such that η o f = φ o ξ. *

3 Topological foldig of locally flat aach spaces 009 ξ f * η V x φ V x W y To complete the defiitio we say that ay map f : S S is a topological foldig. Sice S o cosists of two real umbers,, there are exactly four o topological foldig of S to itself. We deote by F ( M, N ) the set of all topological foldigs of M ito N, ad put F ( M ) = F ( M, M ). If ϕ F ( M, N ), the x M is said to be a sigularity of φ iff φ is ot a local homeomorphism at x. The set of all sigularities of φ is deoted by φ, ad it partitios M ito strata. For each stratum σ, φ σ is a topological immersio oto its image. (3) Foldig of -aach maifolds. A aach maifold M is called a -aach maifold if it is modeled o + the aach space E, for some Z. Theorem (). ϕ F Let ( M, N ) is a discrete subset of M., where M ad N are -aach maifolds. The φ Proof. Let x M ad y = φ ( x ). The there are disc charts ξ : I Vx, η : I Wy o o o M ad N respectively, ad a topological foldig f : S S such that η o f = φ oξ, where = I = [, ]. * o o ξ f * η V x φ V x W y

4 00 E. M. El-Kholy, El-Said R. Lashi ad Salama N. aoud Now suppose that x φ, the f ( ) = f ( ) = ±, say f ( ) = f ( ) =. The f * ( t ) = t. Hece φ is a local homeomorphism o V x { x}. Hece x is a isolated poit of φ, ad so φ is discrete. Theorem (). Let ϕ F ( M, N ). If M ~ R φ is fiite ad # φ is eve., the φ is coutable. If M ~ S, the Proof. The first statemet follows immediately from Theorem (). Suppose that M ~ S, ad let x φ, the there are disc charts ξ : I S, η : I S such that ξ ( 0) = x, η (0) = φ ( x ) = y ad φ oξ = η o f *, where f * : I I is give by f * ( t ) = t. Hece f * iduces orietatios o rays I ( 0 < t < ) ad I + ( < t < 0) ad hece local opposite orietatios o ξ ( I ) ad ξ ( I + ). These local orietatios ca be chose so that each regio has a uique orietatio iduced by disc charts. This shows that the sigularities of φ partitio S ito arcs i such a way that successive arcs have opposite orietatios. Thus the umber of arcs is eve, ad so the umber of sigularities is also eve. (4) Foldig of essetial hypersurfaces A local hypersurface of costat o-zero Riemaia curvature i a locally flat semi-reimaia aach space is a essetial hypersurface (spheres ad pesudospheres resp.) of secod order ad vice versa [3, 4]. Now cosider ay topological foldig ϕ F ( M, N ) where M ad N are essetial hypersurfaces without boudaries ad coected. The disc charts provide local models for the set of sigularities φ, as follows: let φ : S S be a topological foldig. The φ cosists of k poits p,..., p. Hece φ k * cosists of rays joiig each p to 0, that is, φ { : 0 t, i,,..., k } * = = tp i. It follows that the set i φ has the

5 Topological foldig of locally flat aach spaces 0 structure of locally fiite graph eve valecy. A coected subset of p p k 0 K φ embedded i M, for which every vertex has a M \ Kφ is called a φ -regio. p p 3 S f S ) Fig. () It should be oted that the φ -regios together with the edges ad vertices of K φ costitute a topological stratificatio of M. Note also that if M is compact, the K φ is fiite ad the umber of φ -regios is fiite. Moreover, every φ -regio is bouded by a closed polygo i K φ. φ ( (3-) Examples. (a) Let ϕ F ( S ) be a topological foldig of the sphere give by φ ( x, y, z ) = ( x, y, z ). The image of this topological foldig is the orther hemisphere ad φ is the great circle show i Fig. (3). (b) Let Fig. (3) ψ F ( S ) be a topological foldig give by. The image ad ψ are give i Fig. (4). Note ψ ( x, y, z ) = ( x, y, z ) that ψ is a graph.

6 0 E. M. El-Kholy, El-Said R. Lashi ad Salama N. aoud ψ ψ S Fig. (4) ψ ( S ) (c) Let γ F ( P ) be a topological foldig of the pesudosphere which is give by γ ( x, y, z ) = ( x, y, z ). The image γ ( P ) of this foldig ad γ are give i Fig. (5). γ Theorem (3). The composite of ay two topological foldigs of locally flat aach spaces is ot i geeral a topological foldig. We give a example to illustrate this pheomeo. Let φ : S S be give by φ ( x, y, z ) = ( x, y, z ). The ϕ F ( S ), the image of this topological foldig beig the Norther hemisphere H. Let η be a embeddig of the equator z = 0 of S ito S, give by η ( x, y, 0) = x, y, ε xsi, x where 0 < ε <, x 0 ad η (0, y, 0) = (0, y, 0). y the Schoeflies theorem, sice η : H S is a topological embeddig, η exteds to a homeomorphism η : S S. Let ψ = η oφ. The ψ F ( φ o ψ ) has ifiitely may strata. P γ γ ( P ) Fig. (5) ( S ), but ϕoψ F, sice ( S )

7 Topological foldig of locally flat aach spaces 03 We observe that for ay ϕ F ( M, N ) ad for each stratum σ M, φ σ is a topological immersio of σ i N. Suppose ow that ψ F ( N, L) is a topological foldig. The ψ o φ will be a topological foldig if for each stratum σ S, where S is the topological stratificatio iduced by φ o M, φ (σ ) is topologically trasverse to each stratum of ψ. This coditio is ot however, ecessary. The followig theorem shows that the cartezia product of ay two topological foldigs of locally flat aach spaces is a topological foldig. Theorem (4). Let ϕ F ( M, N ) ad ψ F ( K, L ). The ϕ ψ F (, ) M K N L. I this case ( φ ψ ) = ( φ ) K U ( ψ ) K. Proof. First ote that if M ad N are aach maifolds with atlases. = U, φ, E A = U j, φ, E, respectively. The the product { i i i } i I A, { j j } j J M N is a aach maifold with atlas A A = U U, φ φ, E E. { i j i j i j } i I, j J Now let x = ( x, x ) M K, x M, x K ad y = ( y, y ) N L where y N, y L. Let ϕ F ( M, N ), the there are disc chartsξ : Vx for M at x ad η : Wy for N at y ( ) = φ x together with a topological foldig f : S S such that η o f * = φ oξ. f * ξ η V x φ Vx W y Also sice ψ F ( K, L), the there are disc charts ξ : Vx for K at x ad η : W for L at y = ψ ) together with a topological foldig y ( x g : S S such that η o g * = ψ oξ.

8 04 E. M. El-Kholy, El-Said R. Lashi ad Salama N. aoud g * ξ η Thus V x ψ Vx ξ = ξ ξ : Vx = Vx V, : x η = η η Wy W y are disc charts for M K ad N L respectively at x = ( x, x ) ad y = ( φ ψ )( x ) = ( φ ψ )( x, x ) = ( φ ( x ), ψ ( x )) = ( y, y ) respectively with a topological foldig h = f g : S S S S. W y or Thus Now, we have η o f ) ( η o g ) = ( φ oξ ) ( ψ o ( * * ξ ( η η ) o ( f * g* ) = ( φ ψ ) o ( ξ ξ ) η o h = ( φ ψ ) oξ * ξ ) h * = f* g* η Thus V = V V ϕ ψ F ( M K, N L). x x x φ ψ V x W = W W y y y The above theorem ca be geeralized for a fiite umber of topological foldigs. (3-) Examples. (a) Let f F ( I ) ad show i Fig. (6) g F be topological foldigs with images ( S ) Fig. (6) The f G F I S, see Fig. (7) ( )

9 Topological foldig of locally flat aach spaces 05 Fig. (7) (b) Let ϕ F ( S ) ad ψ F ( S ) be topological foldigs give as show i the Fig. (8) Fig. (8) The ϕ ψ F ( S S ) = F ( T ), see Fig. (9) Fig. (9)

10 06 E. M. El-Kholy, El-Said R. Lashi ad Salama N. aoud Refereces [] H. L., Farra, E. El-Kholy, ad S. A. Robertso, Foldig a surface to a polygo, Grometriae edicata 33, pp (55-66), 996. [] E. El-Kholy, ad H. A. Al-Khurasai, Foldig of CW-complex, J. Is. Math. Ad Comp. Sci. (Math. Ser.), Idia, Vol 4, No., pp (4-48), 99. [3] E. R. Lashi ad T. F. Mersal; O hypersurfaces i a locally affie Riemaia aach maifold, Iteratioal Joural of mathematics ad Mathematical Scieces, Vol. 3, Issue 6, pp , 00. [4] E. R. Lashi ad T. F. Mersal: O hypersurfaces i a locally affie Riemaia aach maifold II, Iteratioal Joural of Mathematics ad Mathematical Scieces, i press. [5] S. A. Robertso, Isomatric foldig of Riemaia maifolds, Proc. Roy. Soc. Ediburgh 77, pp. (75-84), 977. [6] S. A Robertso,. ad E El-Kholy,., Ivariat ad equivariat isometric foldig, elta J. Sci., Eqypt, Vol. 8, Vo., pp. ( ), 984. [7] S. A Robertso,. ad E El-Kholy,., Topological foldigs, Commu. Fac. Sci. Uiv. Ak. Ser. Al Vol. 35, pp. (0-07), 986. Received: March, 0