چكيده فارسي :
ﻣﻮﺿﻮﻉ ﻣﻮﺭﺩ ﻣﻄﺎﻟﻌﻪ ﺩﺭ ﺍﯾﻦ ﭘﺎﯾﺎﻥﻧﺎﻣﻪ ﺭﻭﯾﻪﻫﺎﯼ ﺍﻧﺘﻘﺎﻟﯽ ﻭ ﺭﻭﯾﻪﻫﺎﯼ ﻣﺘﺠﺎﻧﺲ ﺑﺎ ﺷﺮﻁ ﺛﺎﺑﺖ ﺑﻮﺩﻥ ﺍﻧﺤﻨﺎﯼ ﮔﺎﻭﺳﯽ ﯾﺎ
ﻣﺘﻮﺳﻂ ﺩﺭ ﻓﻀﺎﯼ ﺍﻗﻠﯿﺪﺳﯽ ﺳﻪ ﺑﻌﺪﯼ ﺍﺳﺖ.ﻫﻤﭽﻨﯿﻦ ﻧﺘﺎﯾﺞ ﺣﺎﺻﻞ ﺩﺭﻣﻮﺭﺩﺭﻭﯾﻪﻫﺎﯼ ﺯﻭﺍﻝﻧﺎﭘﺬﯾﺮﺩﺭﻓﻀﺎﯼ ﻟﻮﺭﻧﺘﺲ‑ﻣﯿﻨﮑﻮﻓﺴﮑﯽ ﺗﻮﺳﻌﻪ ﺩﺍﺩﻩ ﻣﯽﺷﻮﺩ.ﺑﺎﺗﻮﺟﻪ ﺑﻪ ﺗﻌﺮﯾﻒ ﯾﮏ ﺭﻭﯾﻪﯼ ﺍﻧﺘﻘﺎﻟﯽ ﺑﻪ ﺻﻮﺭﺕ ﻣﺠﻤﻮﻉ ﺩﻭ ﺧﻢ ﻣﺜﻞ α ﻭ β، ﺍﯾﻦ ﻣﻄﻠﺐ ﺍﺛﺒﺎﺕ ﺷﺪﻩ ﺍﺳﺖ ﮐﻪ ﺑﺪﻭﻥ ﻫﯿﭻ ﻗﯿﺪﯼ ﺭﻭﯼ α ﻭ β، ﺗﻨﻬﺎ ﺭﻭﯾﻪﻫﺎﯼ ﺍﻧﺘﻘﺎﻟﯽ ﺗﺨﺖ )ﯾﻌﻨﯽ ﺑﺎ ﺍﻧﺤﻨﺎﯼ ﮔﺎﻭﺳﯽ ﺛﺎﺑﺖ ﺻﻔﺮ( ﺭﻭﯾﻪﻫﺎﯼ ﺍﺳﺘﻮﺍﻧﻪﺍﯼ ﻫﺴﺘﻨﺪ.ﺑﻪ ﻋﻼﻭﻩ ﻧﺸﺎﻥ ﺩﺍﺩﻩ ﺷﺪﻩ ﺍﺳﺖ ﮐﻪ ﺍﮔﺮ α ﻭ β ﻣﺴﻄﺢ ﺑﺎﺷﻨﺪ، ﻫﯿﭻ ﺭﻭﯾﻪﯼ ﺍﻧﺘﻘﺎﻟﯽ ﺑﺎ ﺍﻧﺤﻨﺎﯼ ﮔﺎﻭﺳﯽ ﺛﺎﺑﺖ ﻏﯿﺮ ﺻﻔﺮ
K ﻭﺟﻮﺩ ﻧﺪﺍﺭﺩ ﻭ ﺑﺎﻟﻌﮑﺲ. ﺩﺭ ﻣﻮﺭﺩ ﺭﻭﯾﻪﻫﺎﯼ ﻣﺘﺠﺎﻧﺲ ﺍﺛﺒﺎﺕ ﺩﯾﮕﺮﯼ ﺑﺮﺍﯼ ﺍﯾﻦ ﻣﻄﻠﺐ ﺍﺭﺍﺋﻪ ﺷﺪﻩ ﺍﺳﺖ ﮐﻪ ﺻﻔﺤﺎﺕ ﻭ ﭘﯿﭽﻮﺍﺭﻩﻫﺎ ﺗﻨﻬﺎ ﺭﻭﯾﻪﻫﺎﯼ ﻣﺘﺠﺎﻧﺲ ﻣﯿﻨﯿﻤﺎﻝ ﺩﺭ ﻓﻀﺎﯼ ﺍﻗﻠﯿﺪﺳﯽ 3E ﻫﺴﺘﻨﺪ.ﻫﻤﭽﻨﯿﻦ ﺍﯾﻦ ﻣﻄﻠﺐ ﻧﯿﺰ ﺍﺛﺒﺎﺕ ﺷﺪﻩ ﺍﺳﺖ ﮐﻪ ﻫﺮﮔﺎﻩ M ﯾﮏ ﺭﻭﯾﻪﯼ ﻣﺘﺠﺎﻧﺲ ﺩﺭ ﻓﻀﺎﯼ ﺍﻗﻠﯿﺪﺳﯽ 3E ﺑﺎ ﺍﻧﺤﻨﺎﯼ ﮔﺎﻭﺳﯽ ﺛﺎﺑﺖ K ﺑﺎﺷﺪ، ﺁﻧﮕﺎﻩ ﺭﻭﯾﻪ ﺗﺨﺖ ﺍﺳﺖ.ﺑﻪ ﻋﻼﻭﻩ ﺭﻭﯾﻪ ﯾﮏ ﺻﻔﺤﻪ، ﯾﮏ ﺭﻭﯾﻪﯼ ﺍﺳﺘﻮﺍﻧﻪﺍﯼ ﯾﺎ
ﯾﮏ ﺭﻭﯾﻪ ﺑﺎ ﭘﺎﺭﺍﻣﺘﺮﯼﻫﺎﯼ ﺧﺎﺹ ﺑﯿﺎﻥ ﺷﺪﻩ ﺩﺭ ﭘﺎﯾﺎﻥ ﻧﺎﻣﻪ ﺍﺳﺖ. ﺩﺭ ﺣﺎﻟﺖ ﻟﻮﺭﻧﺘﺲ‑ﻣﯿﻨﮑﻮﻓﺴﮑﯽ، ﯾﻌﻨﯽ ﻭﻗﺘﯽ ﮐﻪ 3R ﺑﻪ ﻣﺘﺮ 2−(dz) 2+(dy) 2(dx) ﻣﺠﻬﺰ ﺷﺪﻩ ﺑﺎﺷﺪ، ﯾﮏ ﺭﻭﯾﻪﯼ ﻏﻮﻃﻪﻭﺭ
ﺷﺪﻩ ﺩﺭ 3L ﺭﺍ ﺯﻭﺍﻝﻧﺎﭘﺬﯾﺮ ﮔﻮﯾﻨﺪ،ﺍﮔﺮ ﻣﺘﺮ ﺍﻟﻘﺎ ﺷﺪﻩ ﺭﻭﯼ M ﺯﻭﺍﻝﻧﺎﭘﺬﯾﺮ ﺑﺎﺷﺪ. ﻣﺘﺮ ﺍﻟﻘﺎ ﺷﺪﻩ ﻣﯽﺗﻮﺍﻧﺪ ﺩﻭ ﻧﻮﻉ ﺑﺎﺷﺪ:
ﺍﻟﻒ( ﻣﺜﺒﺖ ﻣﻌﯿﻦ ﻭ ﺩﺭ ﺍﯾﻦ ﺣﺎﻟﺖ ﺭﻭﯾﻪ ﻓﻀﺎﮔﻮﻧﻪ ﻧﺎﻣﯿﺪﻩ ﻣﯽﺷﻮﺩ.
ﺏ( ﻣﺘﺮ ﻟﻮﺭﻧﺘﺴﯽ ﻭ ﺩﺭ ﺍﯾﻦ ﺣﺎﻟﺖ ﺭﻭﯾﻪ ﺯﻣﺎﻥﮔﻮﻧﻪ ﻧﺎﻣﯿﺪﻩ ﻣﯽﺷﻮﺩ.
ﺩﺭ ﻫﺮ ﺩﻭ ﺣﺎﻟﺖ ﺍﺯ ﺭﻭﯾﻪﻫﺎ، ﺍﻧﺤﻨﺎﯼ ﻣﺘﻮﺳﻂ H ﻭ ﺍﻧﺤﻨﺎﯼ ﮔﺎﻭﺳﯽ K ﺑﻪ ﺻﻮﺭﺕ ﺯﯾﺮ ﺑﺮ ﺣﺴﺐ ﻣﺨﺘﺼﺎﺕ ﻣﻮﺿﻌﯽ t) X(s, = X
ﺗﻌﺮﯾﻒ ﻣﯽﺷﻮﺩ:
lG − 2mF + nE ln − m2
H = ϵ2(EG − F 2),K = ϵ EG − F 2
ﺑﻪ ﻗﺴﻤﯽ ﮐﻪ 1− = ϵ ﺍﮔﺮ M ﺯﻣﺎﻥﮔﻮﻧﻪ ﺑﺎﺷﺪ ﻭ 1 = ϵ ﺍﮔﺮ M ﻓﻀﺎﮔﻮﻧﻪ ﺑﺎﺷﺪ. ﻫﻤﭽﻨﯿﻦ G} F, {E, ﻭ n} m, {l, ﺑﻪ
ﺗﺮﺗﯿﺐ ﺿﺮﺍﯾﺐ ﺍﻭﻟﯿﻦ ﻭ ﺩﻭﻣﯿﻦ ﻓﺮﻡ ﺍﺳﺎﺳﯽ ﻧﺴﺒﺖ ﺑﻪ ﻣﺨﺘﺼﺎﺕ ﻣﻮﺿﻌﯽ X ﻫﺴﺘﻨﺪ. ﺩﺭ ﺍﯾﻦ ﭘﺎﯾﺎﻥﻧﺎﻣﻪ، ﻭﺍﺑﺴﺘﮕﯽ ﯾﮏ ﺭﻭﯾﻪ ﺑﺮﺍﯼ ﺍﻧﺘﻘﺎﻟﯽ ﯾﺎ ﻣﺘﺠﺎﻧﺲ ﺑﻮﺩﻥ ﺑﻪ ﺳﺎﺧﺘﺎﺭ ﺁﻓﯿﻦ 3R ﻭ ﺣﺎﺻﻠﻀﺮﺏ ﺗﻮﺍﺑﻊ ﺣﻘﯿﻘﯽ ﺩﺭ R ﻭ ﺍﺳﺘﻘﻼﻝ ﺧﻮﺍﺹ ﺁﻧﻬﺎ ﺍﺯ ﻧﻮﻉ ﻣﺘﺮ، ﺍﺛﺒﺎﺕ ﻧﺘﺎﯾﺞ ﺣﺎﺻﻞ ﺩﺭ ﻓﻀﺎﯼ ﺍﻗﻠﯿﺪﺳﯽ ﺭﺍ ﺩﺭ ﻣﻮﺭﺩ ﺭﻭﯾﻪﻫﺎﯼ ﻭﺍﻗﻊ ﺩﺭ 3L، ﺍﻣﮑﺎﻥﭘﺬﯾﺮ
ﺳﺎﺧﺖ.
چكيده انگليسي :
This M.Sc. thesis is based on [7] and [12]. A translation hypersurface M in Rn+1} can be formulated as
f (x1, ..., xn) = f1(x1) + f2(x2) + + fn(xn), where fi : R R is a smooth functions for 1 i n. In special case, if α : I R R3 and β : J R R3 are smooth curves then a translation surface S is parame- terized by x(s, t) = α(t) + β(s). There are some studies that characterized minimal translation surfaces based on situations of α and β. These curves in details are considered in orthogonal planes or at least one of them assume is a
planner curve. In this thesis, we study on translation surfaces with constant Gaussian curvature. It is proved that the only translation surfaces with zero Gauss curva- ture are cylindrical surfaces. Furthermore, there are no translation surfaces with constant Gaussian curvature K ̸= 0 if one of the generating curves is planar. This means that, we have a complete classification for translation surfaces with constant zero Gaussian curvature and a surface with non zero constant Gaussian curvature cannot be a translation surfaces unless one of the generating curves be planner. We also
deal with another surfaces in this thesis, i.e. the homothetical surfaces. A homothetical hypersurface M in Rn+1}
is defined as graph of a function such f (x1, ..., xn) = f1(x1)f2(x2) . . . fn(xn), where fi : R R is a real smooth functions for 1 i n. Specially, let α : I R R3 and β : J R R3 be smooth curves. Then a translation surface S can be parameterized by x(s, t) = α(t)β(s). This is proved that the only minimal homothet- ical non-degenerate surfaces in L3 are planes and helicoidsstudies that can be characterized based on situations of α and β. We also prove planes and helicoidsstudies are the only minimal homothetical surfaces in three dimensional Eu- clidean space E3. Similar to translation surfaces, we conclude a complete classification for homothetical surfaces with constant Gaussian curvature in E3. In fact it is proved that if S is a homothetical surface in Euclidean space E3 with constant Gaussian curvature K, then K = 0. Furthermore, the surface is a plane, a cylindrical surface or a surface whose parametrization is one of x(s, t) = (s, t, aebs+ct), and x(s, t) = (s, t, ks + d m ft + e 1−m),
where r, b, c > 0, k, f, d, m R, m = 0, 1 and k, f = 0. Then we consider the above mentioned problems in the Lorentzian-Minkowski space L3, that is, R3 endowed with the metric (dx)2 + (dy)2 (dz)2. A surface S in L3 is called nondegenerate if the induced metric on S is not degenerate. The induced metric can only be of two types: positive definite and the surface is called spacelike, or a Lorentzian metric, and the surface is called timelike. In local coordinates x = x(s, t), for both types of surfaces, the mean curvature H and the Gaussian curvature K
are defined and they have expressions K = ϵ ln−m2
EG−F 2
and H = ϵ lG−2mF +nE , where ϵ = 1 if S is spacelike and
2(EG−F 2)
ϵ = 1 if S is timelike. The resultes about translation surface in the Lorentzian-Minkowski space with constant
Gaussian curvature is the same as the results about translation surfaces in E3. Except one case, the mentioned results are tru for homothetical surface in L3. This case state that a minimal homothetical surface S in L3 that defined by x(s, t) = α(t)β(s), is (a part of) a plane or we have one of the following cases:
(i) f (s) = λ1s + λ2 and g(t) = 1 tanh(λ3t + λ4) for λi ∈ R. (ii) f (s) = 1 tanh(λ1s + λ2) and
λ1
g(t) = λ3t + λ4 for λi ∈ R. (iii) f (s) = λ1e
√as and g(t) = λ2e
λ3
at for λ1, λ2, , a ∈ R where a > 0.
