Science Journal Of Mathematics and Statistics
ISSN: 2276-6324
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Research Article
On The Generalized Gaussian and Mean curvatures in E₁ⁿ⁺¹
Ayşe Yavuz, F. Nejat Ekmekci
Department of Secondary Science and Mathematics Education, Necmettin Erbakan University, Konya, Turkey
Department of Mathematics, Faculty of Sciences, Ankara University, , Ankara, Turkey
Abstract:
Before now in [2] the generalized Gaussian and mean curvatures were proved in Euclidean space, but now we prove the theorems in
Lorentzian Space. In our previous paper, we have studied higher order Gaussian curvatures in Lorentzian space. This allowed us to
prove that
𝑛
𝜑(𝑃) = ∏(1 + 𝜀𝑖 𝑟𝑘𝑖 )
𝑖=1
In addition to Gaussian and mean curvatures, 𝐾𝑟 and 𝐻𝑟 for parallel surfaces in 𝐸13 are given. In this study by means of higher order
Gaussian and mean curvatures we calculate the generalized curvatures 𝐾𝑟 and 𝐻𝑟 for parallel surfaces in 𝐸1𝑛+1 .
Keywords: Gaussian curvatures, mean curvatures, parallel hypersurfaces, higher order Gaussian curvatures.
1. Introduction
For an integer 𝑣 with 0 ≤ 𝑣 ≤ 𝑛, changing the first 𝑣 plus signs above to minus gives a metric tensor
𝑣
𝑛
< 𝑣𝑝 , 𝑤𝑝 >= − ∑ 𝑣𝑖 𝑤𝑖 + ∑ 𝑣𝑖 𝑤𝑖
𝑖=1
𝑗=𝑣+1
of index 𝑣 .The resulting semi-Euclidean space 𝑅𝑣𝑛 reduces to 𝑅 𝑛 if 𝑣 = 0. For 𝑛 ≥ 2, 𝑅1𝑛 is called
Minkowski 𝑛 − 𝑠𝑝𝑎𝑐𝑒;if 𝑛= 4 it is the simplest example of a relativistic spacetime.
The common value 𝑣 of index 𝑔𝑝 on a semi-Riemannian manifold 𝑀is a called the index of 0 ≤ 𝑣 ≤
𝑛 = 𝑑𝑖𝑚𝑀 . If 𝑣 = 0, 𝑀is 𝑅𝑖𝑒𝑚𝑎𝑛𝑛𝑖𝑎𝑛 𝑚𝑎𝑛𝑖𝑓𝑜𝑙𝑑; each 𝑔𝑝 is then a (positive definite ) inner product
on 𝑇𝑝 (𝑀). If 𝑣 = 1 𝑎𝑛𝑑 𝑛 ≥ 2, 𝑀 is a Lorentz manifold.
Fix the notation
−1
𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑣,
𝜀𝑖 = {
.
+1 𝑓𝑜𝑟 𝑣 + 1 ≤ 𝑖 ≤ 𝑛
Then the metric of 𝑅𝑣𝑛 can be written
𝑔 = ∑ 𝜀𝑖 𝑑𝑢𝑖 ⊗ 𝑑𝑢𝑖 .
A tangent vector 𝑣 to 𝑀 is
𝑠𝑝𝑎𝑐𝑒𝑙𝑖𝑘𝑒 𝑖𝑓 < 𝑣, 𝑣 > > 0 𝑜𝑟 < 𝑣, 𝑣 >= 0,
𝑛𝑢𝑙𝑙 𝑖𝑓 < 𝑣, 𝑣 >= 0 𝑎𝑛𝑑 𝑣 ≠ 0
𝑡𝑖𝑚𝑒𝑙𝑖𝑘𝑒 𝑖𝑓 < 𝑣, 𝑣 >< 0 [1]
How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "On The Generalized Gaussian and Mean curvatures in E₁ⁿ⁺¹", Science Journal Of Mathematics and
Statistics, Volume 2016, Article ID: sjms-111, 8 Pages, doi: 10.7237/sjms/111
2 | P a g e Science Journal of Mathematics and Statistics (ISSN: 2276-63)
̅ are two hypersurfaces in E₁ⁿ⁺¹ with unit normal vector 𝑁of 𝑀.
Let 𝑀 and 𝑀
𝑁 = ∑𝑛𝑖=1 𝛼𝑖
𝜕
𝜕𝑥 𝑖
̅ such that
where each 𝛼𝑖 is a 𝐶 ∞ function of 𝑀. If there is a function f, from 𝑀 to 𝑀
̅
𝑓: 𝑀 → 𝑀
𝑃 → 𝑓(𝑃) = 𝑃 + 𝑟𝑁𝑃
̅
then 𝑀 is called parallel hypersurfaces of 𝑀, where 𝑟 ∈ 𝑅.[1]
𝑆 denotes the shape operator on M, at 𝑃 ∈ 𝑀.The function 𝐻 defined by
𝐻: 𝑀 → 𝑅
𝑃 → 𝐻(𝑃) = 𝑇𝑟𝑎𝑐𝑒 𝑆(𝑃)
is called the mean curvature function of 𝑀 and the real number 𝐻(𝑃) is called mean curvature of 𝑀 at
the point 𝑃. [3]
The function 𝐾 defined by
𝐾: 𝑀 → 𝑅
𝑃 → 𝐾(𝑃) = 𝜀𝑑𝑒𝑡 𝑆(𝑃)
is called the Gaussian curvature function of 𝑀 and the real number 𝐾(𝑃) is called Gaussian curvature
of 𝑀 at the point 𝑃. [3]
Definition :
Let M be a hypersurfaces in E₁ⁿ⁺¹and 𝑇𝑀 (𝑃) be a tangent space on M, at P∈M. If 𝑆𝑃 denotes the shape
operator on M, then
𝑆𝑃 : 𝑇𝑀 (𝑃) → 𝑇𝑀 (𝑃)
is a linear mapping. If we denote the characteristic vectors by k₁, k₂, ... ,𝑘𝑛 and the corresponding
characteristic vectors by x₁, x₂ ,..., 𝑥𝑛 of 𝑆𝑃 then 𝑘₁, 𝑘₂ , . . . , 𝑘𝑛 are the principal curvatures and x₁, x₂, ...,
𝑥𝑛 are the principal directions of M, at P∈M . On the other hand, if we use the notions 𝜀𝑖 =+1
n
(𝑛)
𝐾1 (k₁, k₂, . . . , 𝑘𝑛 )
= ε1 k1 + ∑ εi k i
i=2
n
(𝑛)
𝐾2 (k₁, k₂, . . . , 𝑘𝑛 )
n
= ∑ εi k i k j + ∑ εi k i k j
i=1<𝑗
i≠1<𝑗
n
(𝑛)
𝐾3 (k₁, k₂, . . . , 𝑘𝑛 )
=
n
∑ εi k i k j k t +
i=1<𝑗<𝑡
∑ εi k i k j k t
i≠1<𝑗<𝑡
⋮
n
(𝑛)
𝐾𝑛 (k1 , k 2 , …
, 𝑘𝑛 ) = ε1 ∏ k i
i=1
then the characteristic polynomial of S(P) becomes
How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "On The Generalized Gaussian and Mean curvatures in E₁ⁿ⁺¹", Science Journal Of Mathematics and
Statistics, Volume 2016, Article ID: sjms-111, 8 Pages, doi: 10.7237/sjms/111
3 | P a g e Science Journal of Mathematics and Statistics (ISSN: 2276-63)
𝑃𝑆(𝑃) (𝑘) = 𝑘ⁿ + (−1)𝐾₁⁽ⁿ⁾𝑘ⁿ⁻¹+. . . +(−1)ⁿ𝐾𝑛 ⁽ⁿ⁾
and K₁,K₂, ... , 𝐾𝑛 are uniquely determined , where the functions 𝐾𝑖 are called the higher ordered
Gaussian curvatures of the hypersurface 𝑀.
Theorem 1:
Let 𝑀 be a hypersurfaces in E₁ⁿ⁺¹ and 𝐾1 , 𝐾2 , . . . , 𝐾𝑛 are called the higher order Gaussian curvatures
and k1 , k 2 , . . . , 𝑘𝑛 are the principal curvatures at the point f(𝑃)∈M. Let us define a function 𝜀𝑖 =+1 and
ε₁=±1 (i≠+1)
𝜑: 𝑀 → 𝑅
𝑃 → 𝜑(𝑃) = 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
𝑛
= ∏(1 + 𝜀𝑖 𝑟𝑘𝑖 )
𝑖=1
such that 𝜑 function is
𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) = 1 + r𝐾1 + 𝑟 2 𝐾2 + ⋯ + 𝑟 𝑛 𝐾𝑛 .
1 ≤ 𝑖 ≤ 𝑛.
Proof:
We prove the theorem by induction method.
a) If 𝑋𝑝 is spacelike ε₁=+1, for 𝑛 = 1, the theorem holds. Actually,
1
𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) = ∏ 1 + 𝜀𝑖 𝑟𝑘𝑖
𝑖=1
= 1 + 𝑟k1
1
= 1 + 𝑟 ∑ εi 𝑘𝑖
i=1
= 1 + 𝑟𝐾1
Now suppose that the theorem holds for 𝑛 − 1 and show that is true for 𝑛:
𝑛−1
𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) = ∏(1 + 𝜀𝑖 𝑟𝑘𝑖 )
𝑖=1
𝑛−1
𝑛−1
𝑛−1
= 1 + 𝑟 ∑ 𝑘𝑖 + 𝑟 2 ∑ 𝑘𝑖 𝑘𝑗 + ⋯ + 𝑟 𝑛−1 ∏ 𝑘𝑖
𝑖=1
𝑖<𝑗
𝑖=1
= 1 + r𝐾1 + 𝑟 2 𝐾2 + ⋯ + 𝑟 𝑛−1 𝐾𝑛−1
For 𝑛, both sides of the equation is multiplied by (1 + 𝑟𝑘𝑛 )
How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "On The Generalized Gaussian and Mean curvatures in E₁ⁿ⁺¹", Science Journal Of Mathematics and
Statistics, Volume 2016, Article ID: sjms-111, 8 Pages, doi: 10.7237/sjms/111
4 | P a g e Science Journal of Mathematics and Statistics (ISSN: 2276-63)
𝑛−1
𝑛−1
𝑛−1
𝑛−1
2
(∏ 1 + 𝑟𝑘𝑖 ) (1 + 𝑟𝑘𝑛 ) = (1 + 𝑟 ∑ 𝑘𝑖 + 𝑟 ∑ 𝑘𝑖 𝑘𝑗 + ⋯ + 𝑟
𝑖=1
𝑖=1
𝑛−1
𝑛−1
∏ 𝑘𝑖 ) (1 + 𝑟𝑘𝑛 )
𝑖<𝑗
𝑛−1
𝑖=1
𝑛−1
𝑛−1
2
𝑛
= 1 + 𝑟 (∑ 𝑘𝑖 + 𝑘𝑛 ) + 𝑟 (∑ 𝑘𝑖 𝑘𝑗 + 𝑘𝑛 ∑ 𝑘𝑖 ) + ⋯ + 𝑟 𝑘𝑛 ∏ 𝑘𝑖
𝑖=1
𝑖<𝑗
𝑖=1
𝑖=1
and we have
𝑛
𝑛
𝑛
𝑛
2
𝑛
∏ 1 + 𝑟𝑘𝑖 = 1 + 𝑟 ∑ 𝑘𝑖 + 𝑟 ∑ 𝑘𝑖 𝑘𝑗 + ⋯ + 𝑟 ∏ 𝑘𝑖
𝑖=1
𝑖=1
𝑖<𝑗
𝑖=1
𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) = 1 + r𝐾1 + 𝑟 2 𝐾2 + ⋯ + 𝑟 𝑛 𝐾𝑛
b) If 𝑋𝑝 is timelike ε₁=-1, for 𝑛 = 1, the theorem holds . Actually,
1
𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) = ∏ 1 + 𝜀𝑖 𝑟𝑘𝑖
𝑖=1
= 1 + 𝜀1 𝑟𝑘1
1
= 1 + 𝑟 ∑ εi 𝑘𝑖
i=1
= 1 + 𝑟𝐾1
Now suppose that the theorem holds for 𝑛 − 1 and show that is true for 𝑛:
𝑛−1
𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) = ∏ 1 + 𝜀𝑖 𝑟𝑘𝑖
𝑖=1
𝑛−1
𝑛−1
𝑛−1
2
= 1 + 𝑟 ∑ 𝜀𝑖 𝑘𝑖 + 𝑟 ∑ 𝜀𝑖 𝑘𝑖 𝑘𝑗 + ⋯ + 𝑟
𝑖=1
𝑛−1
∏ 𝜀𝑖 𝑘𝑖
𝑖<𝑗
𝑖=1
= 1 + r𝐾1 + 𝑟 2 𝐾2 + ⋯ + 𝑟 𝑛−1 𝐾𝑛−1
For 𝑛, both sides of the equation is multiplied by 1 + 𝑟𝜀𝑛 𝑘𝑛
𝑛−1
𝑛−1
𝑛−1
𝑛−1
2
(∏ 1 + 𝑟𝜀𝑖 𝑘𝑖 ) (1 + 𝑟𝜀𝑛 𝑘𝑛 ) = (1 + 𝑟 ∑ 𝜀𝑖 𝑘𝑖 + 𝑟 ∑ 𝜀𝑖 𝑘𝑖 𝑘𝑗 + ⋯ + 𝑟
𝑖=1
𝑖=1
𝑛−1
𝑛−1
𝑖<𝑗
𝑛−1
∏ 𝜀𝑖 𝑘𝑖 ) (1 + 𝑟𝜀𝑛 𝑘𝑛 )
𝑖=1
𝑛−1
2
𝑛−1
𝑛
= 1 + 𝑟 ( ∑ 𝜀𝑖 𝑘𝑖 + 𝜀𝑛 𝑘𝑛 ) + 𝑟 (∑ 𝜀𝑖 𝑘𝑖 𝑘𝑗 + 𝑘𝑛 ∑ 𝜀𝑖 𝑘𝑖 ) + ⋯ + 𝑟 𝜀𝑛 𝑘𝑛 ∏ 𝜀𝑖 𝑘𝑖
𝑖=1
𝑖<𝑗
𝑖=1
𝑖=1
and we have
How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "On The Generalized Gaussian and Mean curvatures in E₁ⁿ⁺¹", Science Journal Of Mathematics and
Statistics, Volume 2016, Article ID: sjms-111, 8 Pages, doi: 10.7237/sjms/111
5 | P a g e Science Journal of Mathematics and Statistics (ISSN: 2276-63)
𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) = 1 + r𝐾1 + 𝑟 2 𝐾2 + ⋯ + 𝑟 𝑛 𝐾𝑛
Theorem 2:
Let 𝑀 be a hypersurfaces in E₁ⁿ⁺¹ and 𝐾1 , 𝐾2 , . . . , 𝐾𝑛 are called the higher order Gaussian curvatures
̅ and 𝐻
̅ are generalized Gaussian
and k1 , k 2 , . . . , 𝑘𝑛 are the principal curvatures at the point f(𝑃)∈M. 𝐾
̅ at the point 𝑓(𝑃). Suppose the function
and mean curvatures of 𝑀
𝜑: 𝑀 → 𝑅
𝑃 → 𝜑(𝑃) = 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
𝑛
= ∏(1 + 𝜀𝑖 𝑟𝑘𝑖 )
𝑖=1
such that 𝜀𝑖 =+1 and ε₁=±1 (i≠+1)
Then we have
𝜕 𝑛 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
𝜕𝑟 𝑛
̅=𝜀
𝐾
𝑛! 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
and
𝜕𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
1
𝜕𝑟
̅=
𝐻
𝑛 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
Proof: If
is principal curvatures of
̅̅̅at the point
curvatures of 𝑀
at the point
in direction
preserves principal directions, where
in direction
, then
is the principal
that is,
which means that
is the differential of
and we know that
then we know that the shape operator of ̅̅̅
𝑀 is
𝜀1 𝑘1
1 + 𝑟𝜀1 𝑘1
𝑆𝒓 =
⋮
[
0
⋯
0
⋱
⋮
𝜀𝑛 𝑘𝑛
⋯
1 + 𝑟𝜀𝑛 𝑘𝑛 ]
and
How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "On The Generalized Gaussian and Mean curvatures in E₁ⁿ⁺¹", Science Journal Of Mathematics and
Statistics, Volume 2016, Article ID: sjms-111, 8 Pages, doi: 10.7237/sjms/111
6 | P a g e Science Journal of Mathematics and Statistics (ISSN: 2276-63)
̅ = det 𝑆𝑟
𝐾
=𝜀 (
𝜀1 𝑘1
1+𝑟𝜀1 𝑘1
=𝜀
…
𝜀𝑛 𝑘𝑛
1+𝑟𝜀𝑛 𝑘𝑛
)
𝜀1 𝑘1 𝜀2 𝑘2 …𝜀𝑛 𝑘𝑛
∏𝑛
𝑖=1(1+𝜀𝑖 𝑟𝑘𝑖 )
=𝜀
∏𝑛
𝑖=1 𝜀𝑖 𝑘𝑖
∏𝑛
𝑖=1(1+𝜀𝑖 𝑟𝑘𝑖 )
We multiply the right sides of the equation with 𝑛!
𝑛! ∏𝑛𝑖=1 𝜀𝑖 𝑘𝑖
=𝜀
𝑛! ∏𝑛𝑖=1(1 + 𝜀𝑖 𝑟𝑘𝑖 )
𝑛! 𝐾𝑛
=𝜀
𝑛! 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
and we derivate to 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) order n according to r
𝜕𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) 𝜕(1 + r𝐾1 + 𝑟 2 𝐾2 + ⋯ + 𝑟 𝑛 𝐾𝑛 )
=
𝜕𝑟
𝜕𝑟
=𝐾1 + 2𝑟𝐾2 + ⋯ + 𝑛𝑟 𝑛−1 𝐾𝑛
𝜕 2 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) 𝜕(𝐾1 + 2𝑟𝐾2 + ⋯ + 𝑛𝑟 𝑛−1 𝐾𝑛 )
=
𝜕𝑟 2
𝜕𝑟
= 2𝐾2 + ⋯ + 𝑛(𝑛 − 1)𝑟 𝑛−2 𝐾𝑛
and we continue to derivation, we have
𝜕 𝑛 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
= 𝑛! 𝐾𝑛
𝜕𝑟 𝑛
and we obtain with implying equality
𝜕 𝑛 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
𝜕𝑟 𝑛
̅=𝜀
𝐾
.
𝑛! 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
We proof the other equality
̅=
𝐻
=
=
1
𝐼𝑧 𝑆𝑟
𝑛
1
𝜀1 𝑘1
𝜀𝑛 𝑘𝑛
(
+ ⋯+
)
𝑛 1 + 𝑟𝜀1 𝑘1
1 + 𝑟𝜀𝑛 𝑘𝑛
1 𝜀1 𝑘1 ∏𝑛𝑖=2(1 + 𝜀𝑖 𝑟𝑘𝑖 ) + 𝜀2 𝑘2 (1 + 𝜀1 𝑟𝑘1 ) ∏𝑛𝑖=3(1 + 𝜀𝑖 𝑟𝑘𝑖 ) + ⋯ + 𝜀𝑛 𝑘𝑛 ∏𝑛−1
𝑖=1 (𝜀𝑖 𝑟𝑘𝑖 )
(
)
𝑛
∏𝑖=1(1 + 𝜀𝑖 𝑟𝑘𝑖 )
𝑛
How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "On The Generalized Gaussian and Mean curvatures in E₁ⁿ⁺¹", Science Journal Of Mathematics and
Statistics, Volume 2016, Article ID: sjms-111, 8 Pages, doi: 10.7237/sjms/111
7 | P a g e Science Journal of Mathematics and Statistics (ISSN: 2276-63)
We derivate according to r
𝜕𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 ) 𝜕(∏𝑛𝑖=1(1 + 𝜀𝑖 𝑟𝑘𝑖 ))
=
𝜕𝑟
𝜕𝑟
=
𝜕((1 + 𝜀1 𝑟𝑘1 )(1 + 𝜀2 𝑟𝑘2 ) … (1 + 𝜀𝑛 𝑟𝑘𝑛 ))
𝜕𝑟
= 𝜀1 𝑘1 (1 + 𝜀2 𝑟𝑘2 )(1 + 𝜀3 𝑟𝑘3 ) … (1 + 𝜀𝑛 𝑟𝑘𝑛 )
+(1 + 𝜀1 𝑟𝑘1 )𝜀2 𝑘2 (1 + 𝜀3 𝑟𝑘3 ) … (1 + 𝜀𝑛 𝑟𝑘𝑛 )
+(1 + 𝜀1 𝑟𝑘1 )(1 + 𝜀2 𝑟𝑘2 )𝜀3 𝑘3 … (1 + 𝜀𝑛 𝑟𝑘𝑛 )
⋯
+(1 + 𝜀1 𝑟𝑘1 )(1 + 𝜀2 𝑟𝑘2 )(1 + 𝜀3 𝑟𝑘3 ) … (1 + 𝜀𝑛−1 𝑟𝑘𝑛−1 )𝜀𝑛 𝑘𝑛
𝑛
𝑛
𝑛−1
= 𝜀1 𝑘1 ∏(1 + 𝜀𝑖 𝑟𝑘𝑖 ) + 𝜀2 𝑘2 (1 + 𝜀1 𝑟𝑘1 ) ∏(1 + 𝜀𝑖 𝑟𝑘𝑖 ) + ⋯ + 𝜀𝑛 𝑘𝑛 ∏(𝜀𝑖 𝑟𝑘𝑖 )
𝑖=2
𝑖=3
𝑖=1
So we have last equation and we obtain that
𝜕𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
1
𝜕𝑟
̅=
𝐻
.
𝑛 𝜑(r, k1 , k 2 , . . . , 𝑘𝑛 )
References
[1]O.Neill , B., Semi Riemannian Geometry. Department of Mathematics University of California Los Angeles, California. 1983
[2]Sağel M.K.and Hacısalihoğlu, H.H.1988. On the Gaussian and mean curvatures of a paralel hypersurface I: Commun. Fac. Sci.Univ. Ankara, Ser. Al
37,No.1-2,9-1500.
[3] Yaşar, A. Higher Order Gaussian Curvatures of a Parallel Hypersurfaces in
Ln Lorentz Space, Master Thesis Ankara University.
How to Cite this Article: Ayşe Yavuz, F. Nejat Ekmekci, "On The Generalized Gaussian and Mean curvatures in E₁ⁿ⁺¹", Science Journal Of Mathematics and
Statistics, Volume 2016, Article ID: sjms-111, 8 Pages, doi: 10.7237/sjms/111