On The Generalized Gaussian and Mean curvatures in E₁ⁿ+¹. Ayşe Yavuz, F. Nejat Ekmekci

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1 Sciece Joural Of Mathematics ad Statistics ISSN: Author(s) 206. CC Attributio 3.0 Licese. Published By Sciece Joural Publicatio Iteratioal Ope Access Publisher Research Article O The Geeralized Gaussia ad Mea curvatures i E₁ⁿ+¹ Ayşe Yavuz, F. Nejat Ekmekci Departmet of Secodary Sciece ad Mathematics Educatio, Necmetti Erbaka Uiversity, Koya, Turkey Departmet of Mathematics, Faculty of Scieces, Akara Uiversity,, Akara, Turkey Abstract: Before ow i [2] the geeralized Gaussia ad mea curvatures were proved i Euclidea space, but ow we prove the theorems i Loretzia Space. I our previous paper, we have studied higher order Gaussia curvatures i Loretzia space. This allowed us to prove that φ(p) = ( + ε i r ) I additio to Gaussia ad mea curvatures, K r ad H r for parallel surfaces i E 3 are give. I this study by meas of higher order Gaussia ad mea curvatures we calculate the geeralized curvatures K r ad H r for parallel surfaces i E +. Keywords: Gaussia curvatures, mea curvatures, parallel hypersurfaces, higher order Gaussia curvatures.. Itroductio For a iteger v with 0 v, chagig the first v plus sigs above to mius gives a metric tesor v < v p, w p >= v i w i + v i w i j=v+ of idex v.the resultig semi-euclidea space R v reduces to R if v = 0. For 2, R is called Mikowski space;if = 4 it is the simplest example of a relativistic spacetime. The commo value v of idex g p o a semi-riemaia maifold Mis a called the idex of 0 v = dimm. If v = 0, Mis Riemaia maifold; each g p is the a (positive defiite ) ier product o T p (M). If v = ad 2, M is a Loretz maifold. Fix the otatio ε i = { for i v, + for v + i. The the metric of R v ca be writte A taget vector v to M is g = ε i du i du i. spacelike if < v, v > > 0 or < v, v >= 0, ull if < v, v >= 0 ad v 0 timelike if < v, v >< 0 [] How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "O The Geeralized Gaussia ad Mea curvatures i E₁ⁿ+¹", Sciece Joural Of Mathematics ad

2 2 P a g e Sciece Joural of Mathematics ad Statistics (ISSN: ) Let M ad M are two hypersurfaces i E₁ⁿ+¹ with uit ormal vector Nof M. N = α i where each α i is a C fuctio of M. If there is a fuctio f, from M to M such that f: M M P f(p) = P + rn P the M is called parallel hypersurfaces of M, where r R.[] S deotes the shape operator o M, at P M.The fuctio H defied by H: M R xi P H(P) = Trace S(P) is called the mea curvature fuctio of M ad the real umber H(P) is called mea curvature of M at the poit P. [3] The fuctio K defied by K: M R P K(P) = εdet S(P) is called the Gaussia curvature fuctio of M ad the real umber K(P) is called Gaussia curvature of M at the poit P. [3] Defiitio : Let M be a hypersurfaces i E₁ⁿ+¹ad T M (P) be a taget space o M, at P M. If S P deotes the shape operator o M, the S P : T M (P) T M (P) is a liear mappig. If we deote the characteristic vectors by k₁, k₂,...,k ad the correspodig characteristic vectors by x₁, x₂,..., x of S P the k₁, k₂,..., k are the pricipal curvatures ad x₁, x₂,..., x are the pricipal directios of M, at P M. O the other had, if we use the otios ε i =+ K () (k₁, k₂,..., k ) = ε k + ε i i=2 K () 2 (k₁, k₂,..., k ) = ε i k j + ε i k j <j i <j K () 3 (k₁, k₂,..., k ) = ε i k j k t + ε i k j <j<t i <j<t K () (k, k 2,, k ) = ε the the characteristic polyomial of S(P) becomes k t How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "O The Geeralized Gaussia ad Mea curvatures i E₁ⁿ+¹", Sciece Joural Of Mathematics ad

3 3 P a g e Sciece Joural of Mathematics ad Statistics (ISSN: ) P S(P) (k) = kⁿ + ( )K₁(ⁿ)kⁿ ¹+... +( )ⁿK (ⁿ) ad K₁,K₂,..., K are uiquely determied, where the fuctios K i are called the higher ordered Gaussia curvatures of the hypersurface M. Theorem : Let M be a hypersurfaces i E₁ⁿ+¹ ad K, K 2,..., K are called the higher order Gaussia curvatures ad k, k 2,..., k are the pricipal curvatures at the poit f(p) M. Let us defie a fuctio ε i =+ ad ε₁=± (i +) φ: M R such that φ fuctio is P φ(p) = φ(r, k, k 2,..., k ) = ( + ε i r ) φ(r, k, k 2,..., k ) = + rk + r 2 K r K. i. Proof: We prove the theorem by iductio method. a) If X p is spacelike ε₁=+, for =, the theorem holds. Actually, φ(r, k, k 2,..., k ) = + ε i r = + rk = + r ε i = + rk Now suppose that the theorem holds for ad show that is true for : φ(r, k, k 2,..., k ) = ( + ε i r ) = + r + r 2 k j + + r = + rk + r 2 K r K For, both sides of the equatio is multiplied by ( + rk ) How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "O The Geeralized Gaussia ad Mea curvatures i E₁ⁿ+¹", Sciece Joural Of Mathematics ad

4 4 P a g e Sciece Joural of Mathematics ad Statistics (ISSN: ) ad we have ( + r ) ( + rk ) = ( + r + r 2 k j + + r ) ( + rk ) = + r ( + k ) + r 2 ( k j + k ) + + r k + r = + r + r 2 k j + + r φ(r, k, k 2,..., k ) = + rk + r 2 K r K b) If X p is timelike ε₁=-, for =, the theorem holds. Actually, φ(r, k, k 2,..., k ) = + ε i r = + ε rk = + r ε i = + rk Now suppose that the theorem holds for ad show that is true for : φ(r, k, k 2,..., k ) = + ε i r = + r ε i + r 2 ε i k j + + r ε i = + rk + r 2 K r K For, both sides of the equatio is multiplied by + rε k ( + rε i ) ( + rε k ) = ( + r ε i + r 2 ε i k j + + r ε i ) ( + rε k ) ad we have = + r ( ε i + ε k ) + r 2 ( ε i k j + k ε i ) + + r ε k ε i How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "O The Geeralized Gaussia ad Mea curvatures i E₁ⁿ+¹", Sciece Joural Of Mathematics ad

5 5 P a g e Sciece Joural of Mathematics ad Statistics (ISSN: ) φ(r, k, k 2,..., k ) = + rk + r 2 K r K Theorem 2: Let M be a hypersurfaces i E₁ⁿ+¹ ad K, K 2,..., K are called the higher order Gaussia curvatures ad k, k 2,..., k are the pricipal curvatures at the poit f(p) M. K ad H are geeralized Gaussia ad mea curvatures of M at the poit f(p). Suppose the fuctio φ: M R such that ε i =+ ad ε₁=± (i +) The we have ad P φ(p) = φ(r, k, k 2,..., k ) K = ε = ( + ε i r ) φ(r, k, k 2,..., k )! φ(r, k, k 2,..., k ) H = φ(r, k, k 2,..., k ) φ(r, k, k 2,..., k ) Proof: If is pricipal curvatures of at the poit i directio, the is the pricipal curvatures of M at the poit i directio that is, which meas that preserves pricipal directios, where is the differetial of ad we kow that the we kow that the shape operator of is M ad S r = ε k + rε k 0 ε k 0 [ + rε k ] How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "O The Geeralized Gaussia ad Mea curvatures i E₁ⁿ+¹", Sciece Joural Of Mathematics ad

6 6 P a g e Sciece Joural of Mathematics ad Statistics (ISSN: ) =ε ( ε k K = det S r ε k ) +rε k +rε k =ε ε k ε 2 k 2 ε k = ε (+ε i r ) ε i (+ε i r ) We multiply the right sides of the equatio with!! ε i = ε! ( + ε i r )! K = ε! φ(r, k, k 2,..., k ) ad we derivate to φ(r, k, k 2,..., k ) order accordig to r φ(r, k, k 2,..., k ) ad we cotiue to derivatio, we have ad we obtai with implyig equality We proof the other equality = ( + rk + r 2 K r K ) =K + 2rK r K 2 φ(r, k, k 2,..., k ) 2 = (K + 2rK r K ) = 2K ( )r 2 K φ(r, k, k 2,..., k ) =! K K = ε φ(r, k, k 2,..., k )! φ(r, k, k 2,..., k ). H = Iz S r = ( ε k + rε k + + ε k + rε k ) = (ε k i=2( + ε i r ) + ε 2 k 2 ( + ε rk ) i=3 ( + ε i r ) + + ε k (ε i r ) ) ( + ε i r ) How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "O The Geeralized Gaussia ad Mea curvatures i E₁ⁿ+¹", Sciece Joural Of Mathematics ad

7 7 P a g e Sciece Joural of Mathematics ad Statistics (ISSN: ) We derivate accordig to r φ(r, k, k 2,..., k ) = ( ( + ε ir )) = (( + ε rk )( + ε 2 rk 2 ) ( + ε rk )) = ε k ( + ε 2 rk 2 )( + ε 3 rk 3 ) ( + ε rk ) +( + ε rk )ε 2 k 2 ( + ε 3 rk 3 ) ( + ε rk ) +( + ε rk )( + ε 2 rk 2 )ε 3 k 3 ( + ε rk ) +( + ε rk )( + ε 2 rk 2 )( + ε 3 rk 3 ) ( + ε rk )ε k = ε k ( + ε i r ) + ε 2 k 2 ( + ε rk ) ( + ε i r ) + + ε k (ε i r ) i=2 So we have last equatio ad we obtai that i=3 H = φ(r, k, k 2,..., k ) φ(r, k, k 2,..., k ). Refereces []O.Neill, B., Semi Riemaia Geometry. Departmet of Mathematics Uiversity of Califoria Los Ageles, Califoria. 983 [2]Sağel M.K.ad Hacısalihoğlu, H.H.988. O the Gaussia ad mea curvatures of a paralel hypersurface I: Commu. Fac. Sci.Uiv. Akara, Ser. Al 37,No.-2, [3] Yaşar, A. Higher Order Gaussia Curvatures of a Parallel Hypersurfaces i L Loretz Space, Master Thesis Akara Uiversity. How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "O The Geeralized Gaussia ad Mea curvatures i E₁ⁿ+¹", Sciece Joural Of Mathematics ad