A Thesis in Applied Mathematics Surface of second degree in 3-Minkowski space b Sun Yan Supervisor: Professor Liu Huili Northeastern Universit Decembe

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2 A Thesis in Applied Mathematics Surface of second degree in 3-Minkowski space b Sun Yan Supervisor: Professor Liu Huili Northeastern Universit December 3

3 : - I -

4 Minkowski Minkowski Euclid Minkowski Minkowski Minkowski Minkowski Minkowski, - II -

5 ABSTRACT Surface of second degree in 3-Minkowski space ABSTRACT As a new field, the Minkowski space is alwas the focus of Mathematics and Phsics. As we know, there are man meaningful results for curves and surfaces in the Euclidean space. The Minkowski space is a linear space with indefinite inner product. So the curves and the surfaces in the Minkowski space are ver different as in the Euclidean space. In this paper, we will show a method to stud surface of second degree in three Minkowski space. The main works are as follows: At first, we stud the distance formulas for point, straight line and plane in three dimensional Minkowski space. Then, with the relations between the point and the straight line, the point and the plane, the straight line and the straight line, we get a lot of surfaces of second degree. Finall, the surfaces will be simplified with the knowledge of algebra and some special traces are given. Ke words: Minkowski space, surface of second degree, distance formula, indefinite inner product - III -

6 ...I... II ABSTRACT...III...IV Minkowski Minkowski Minkowski...8 Minkowski IV -

8 - - -Minkowski Euclid Minkowski

9 . Euclid []-[4] [5]-[3] [9] Minkowski [4]-[8] Euclid Minkowski Minkowski Minkowski - 3 -

10 Euclid. 3.. E 3 E ur α β ur E 3 ur α β ur ur ur αβ, = x x x 3 3 { x, x, x } {,, } ur α β ur E ur α β ur ur ur αβ, = x x x Minkowski E ds = dx dx dx3 ur ur ur ur... αβ, E 3 3 ur ur αβ, = x xx33 ur urr ur 3... αβγ,, E { α = x, x, x } r γ = { z, z, z } 3 ur ur α, β ur ur α β α = { x, x, x } β = {,, } 3 x x x x x x,, 3 3 = 3 3 ur 3 β = {,, } 3 ur ur α, β

11 ur ur ur ur r β γ = ( α ) x x x z z z ur ur ur ur α β, α β λα µβ ( λ, µ ) 3.. E ur ur ur 3... α E αα, > ur ur ur ur αα, < αα, = 3..3 E r r v = { l, m, n} L v E rr vv, > L rr vv, < L rr vv, = L 3..4 E P x z P xz,, uuur r uuur r n= { A, B, C} PP, n =..4. (,, ) PP Ax B Cz D= D =Ax B Cz r n= A, B, C { } - 5 -

12 r..4. π : Ax B Cz D = n = A, B, C { } r r nn, > r r nn, < r r nn, =... P x,, z π : Ax B Cz D = d = Ax B Cz D A B C.. P x,, z L : x x z = = z l m n d = z z z z x x x x m n n l l m l m n..3 : x x L z = = z l m n L : x x z = = z l m n - 6 -

13 d = x x z z l m n l m n m n n l l m m n n l l m - 7 -

14 Minkowski Minkowski Minkowski 3. E E () E ur uur () v = x, x, x v =,, 3.. { } 3 x x x = = 3 = 3 x x x = a 3 3 = < v uur ur uur () v = x, x, x v =,, { } a t t { } 3 { } 3 ()

15 a = t = 3 v ur Minkowski 3..3 P( x,, z ) π : Ax B Cz D = P( x,, z ) π : Ax B Cz D= d = Ax B Cz D ε ( A B C ) (3..) ε =± P ( x,, z ) P( x,, z) r uuuur { n= A, B, C} PP 3.. xx zz = = = t A B C (3..) (3..) (3..3) Ax B Cz D = (3..3) t = Ax B Cz D A B C x x = A( Ax B Cz D) A B C = BAx ( B Cz D) A B C C( Ax B Cz D) z z = A B C (3..4) () π n r uuuur 3..PP - 9 -

16 d = uuuuruuuur PP, PP Minkowski ( x x ) ( ) ( z z ) (3..4) (3..5) = (3..5) d = Ax B Cz D A B C () π n r uuuur 3..PP d = uuuuruuuur PP, PP ( z z ) ( x x ) ( ) = (3..6) (3..4) (3..6) d = Ax B Cz D C A B P( x,, z) L : x x z z = = l m n P ( x,, z ) L : x x z z = = l m n d = ε x x z z z z x x l m m n n l ε ( l m n ) ε =±, i =, i P ( x,, z ) L P( x,, z) L r uuuur v = l, m, n { PP = x x,, z z } { } - -

17 r uuuur vpp, = Minkowski 3.. lx ( x) m ( ) nz ( z) = (3..) x x z z = = = t l m n x = x lt = mt (3..) (3..) x z x = z = z nt. (3..) lx ( x) m ( ) nz ( z) t = l m n ( n m )( x x) l m( ) l n( zz) l m n l m x x n l m n z z = l m n l n x x m n l m z z z = l m n (3..3) () L v r uuuur PP P( x,, z ) L d = uuuuruuuur PP, PP ( x x ) ( ) ( z z ) = (3..4) - -

18 Minkowski (3..3) (3..4) d = z z z z x x x x m n n l l m n l m () L v r uuuur PP uuuur PP P( x,, z) L d = uuuuruuuur PP, PP ( x x ) ( ) ( z z ) (3..3) (3..5)) = (3..5) d = x x z z z z x x l m m n n l l m n uuuur PP d = uuuuruuuur PP, PP ( z z ) ( x x ) ( ) = (3..6) (3..3) (3..6) d = z z z z x x x x m n n l l m l m n

19 : x x L z = = z l m n L : x x z = = z l m n Minkowski d = ε x x z z l m n l m n m n n l l m ε m n n l l m x x z z ε =± ε = ± ε l m n l m n L, L, P x,, z, P x,, z PP L, L 3 4 x3 = x lt 3 = mt L L z3 = z nt (3.3.) x4 = x lt 4 = mt z4 = z nt (3.3.) uuuur PP = x x z z {,, } (3.3.3) - 3 -

20 Minkowski uuuur r uuuur r PP 3 4 v 3 P P4 v (3.3.)(3.3.)(3.3.3) t = l( nl nl) m( nm nm) ( z z) ( nm nm) ( nl nl ) ( lm l m) t n( nl nl) m( lm lm) ( x x) ( nm nm) ( nl nl ) ( lm l m) l( lm lm) n( nm nm) ( ) ( nm nm) ( nl nl ) ( lm l m) = l( nl nl) m( nm nm) ( z z) ( nm nm) ( nl nl ) ( lm l m) n( nl nl) m( lm lm) ( x x) ( nm nm) ( nl nl ) ( lm l m) l( lm lm) n( nm nm) ( ) ( nm nm) ( nl nl ) ( lm l m) x x z z ( x x ) ( ) ( z z ) = x x z z m n n l l m l m n m n n l l m l m n m n n l l m m n n l l m (3.3.4) m n n l l m v r v r m n n l l m v, v ur uur { v = l, m, n}, v = { l, m, n } L, L uuuur r r () L L P P v v 3 4

21 Minkowski uuuur PP 3 4 d = uuuur uuuur PP, PP ( x x ) ( ) ( z z ) = (3.3.5) (3.3.4) (3.3.5) d = ε x x z z l m n l m n m n n l l m m n n l l m (3.3.6) x x z z ε =± l m n l m n () L L L L uuuur PP 3 4 L, L (3.3.6) (3) L L v r v r uuuur 3 PP 4 uuuur PP 3 4 L, L (3.3.6) uuuur r r PP 3 4 v v d = uuuur uuuur PP, PP ( z z ) ( x x ) ( ) = (3.3.7) (3.3.4) (3.3.7) - 5 -

22 d = x x z z ε l m n l m n l m m n n l l m m n n l x x z z ε =± l m n l m n Minkowski - 6 -

23 Minkowski Minkowski,, Minkowski V F x, z, V, kl F V B: V V F ( ) Bkx ( lz, ) = kb( xz, ) lb( z, ) ( ) B( x, k lz) = kb( x, ) lb( x, z) B V (, ) B(, x) B x = x, V B V { } V n- V e, e,..., e V n B F n B = b ij B ( x, ) = i, j b ij x i x j j i x = xe, = e i j 4..,..., n x x (,..., ) f x x = b x x n i j ij i, j - 7 -

24 Minkowski F b F b = b B ij b b... b b b... b.. b b... b n n = n n nn ij V q e, e,..., e n x,..., n x x ( n,.., ) ji T q x = f x x = x Bx { } F n f x,.., x x = P 4. Minkowski 4.. Minkowski 4... A,B n A B a a... a n an a a... an an A =... an an... an n an n an an... ann a nn b b... b n b n b b... b n b n B =... bn bn... bn n bn n bn bn... bnn b nn

25 A B = n n n a b a b a b a b... a b a b Minkowski i i n n i i n n i in n nn i= i= i= n n n a b a b a b a b... a b a b i i n n i i n n i in n nn i= i= i= n n n... ab ab ab ab... ab ab ni i nn n ni i nn n ni in nn nn i= i= i= 4.. Minkowski { } 4 Minkowski ξ = x,, z, t R 4 { x,, z, t } R η = ( ξ, η ) = B x x z z tt 4 4 B: R R R Minkowski n f x, x,..., x = a x a xx a xx... a xx n 3 3 n n a x a x x a x x... annxn a a... a n a n a a... an an A =... an an... an n an n an an... ann a nn 4..3 Minkowski Minkowski

26 Minkowski a A n n n- n- n- n- n n- n a = 4.. Minkowski n 4..4 Minkowski (,, ) = f x x x a x a x x a x x a x a x x a x A a x a x a x a (,, ) = g x x x a x a x x a x x a x a x x a x a a a A= a a a a a a A a a a a A a a a (4..) - -

27 A A Minkowski a a a a a a a a a a a a a aa a f x, x, x (4..) 3 (4..) a x a x a x a x a x a x a = '' '' '' a x a x a x a ' ' ' ' ' ' ' = - -

28 Minkowski Minkowski 5. P xz,, π : Ax B Cz D= P x,, z l l > 5.. Minkowski uuur π PP Ax B Cz D A B C ( x x ) ( ) ( zz ) = l ( Ax B Cz D) l ( A B C ) ( x x ) ( ) ( z z ) = A A B C x B A B C l l - -

29 Minkowski C l A B C z ABx BCz ACxz l l ( ) AD A B C x x BD A B C CD l A B C z z D l A B C x z = (5..) uuur π PP Ax B Cz D A B C ( zz ) ( xx ) ( ) = l ( Ax B Cz D) = l ( A B C ) ( z z ) ( x x ) ( ) A A B C x B A B C l l C l A B C z ABx BCz ACxz AD A B C x x BD A B C l l ( ) CD A B C z z D A B C x z = l l (5..) uuur π PP Ax B Cz D C A B ( x x ) ( ) ( zz ) = l ( Ax B Cz D) = l ( C A B ) ( x x ) ( ) ( z z ) - 3 -

30 Minkowski A A B C x B A B C l l C l A B C z ABx BCz ACxz AD A B C x x BD A B C l l ( ) CD A B C z z D A B C x z = l l (5..) uuur π PP Ax B Cz D C A B ( zz ) ( xx ) ( ) = l ( Ax B Cz D) = l ( C A B ) ( z z ) ( x x ) ( ) A A B C x B A B C l l C l A B C z ABx BCz ACxz l l ( ) AD A B C x x BD A B C CD l A B C z z D l A B C x z = (5..) 5.. (5..) (5..) Minkwoski x, zxz, - 4 -

31 Minkowski. (5..) (5..) A l A B C AB AC AB B l A B C BC AC BC C l A B C (4..) Τ Τ Τ3 (5..3) Τ = A l A B C l Τ = ( A B C ) l ( A B C ) ( A B ) A l ( A B C ) ( l )( A B C ) ( A B C ) ( A B ) l Τ = l 3 (4..) AB AC A l A B C A B C A B BC l l ( A B C ) ( A B ) (5..4)

32 Minkowski (5..) ( A B C ) l ( A B C ) ( A B ) A l ( A B C ) ( l )( A B C ) z D ( A B C ) ( A B ) l A ( A B C ) x l l = l D A, BCx,,,, z. (5..) (5..) A A B C AB AC l AB B A B C BC l AC BC C l A B C (4..) (5..5) T T T 3 Τ = A l A B C l Τ = ( A B C ) l ( A B C ) ( A B ) A l ( A B C ) l Τ = l ( l )( A B C ) ( A B C ) ( A B ) 3-6 -

33 Minkowski (4..) AB AC A l A B C A B C A B BC l l (5..) ( A B C ) ( A B ) ( A B C ) l ( A B C ) ( A B ) A l ( A B C ) ( l )( A B C ) z D ( A B C ) ( A B ) l A ( A B C ) x l l = l D A, BCx,,,, z 5..3 (5..6) (5..5) (5..6) λ. (5..5) A, B, C l ( A B C ) l ( A B C ) ( A B ) l ( ) A l ( A B C ) ( l )( A B C ) ( A B C ) ( A B ) A A B C l l (),, () A A B C < l < B C < - 7 -

34 Minkowski A B l A B C < () l A B C Ax B Cz D = (3) A B l > A B C Ax B Cz D = (4) A B A B C < l A A B < l < A B C A B C Ax B Cz D = (5) A A B C < l - 8 -

35 Minkowski Ax B Cz D = (6) A l A B C Ax B Cz D = () () l > () < l <. (5..6) A, B, C Ax B Cz D = Ax B Cz D = l ( A B C ) l ( A B C ) ( A B ) l ( ) A l ( A B C ) ( l )( A B C ) ( A B C ) ( A B ) A A B C l l (),, Ax B Cz D = () () l < A A B C - 9 -

36 Minkowski Ax B Cz D = () l > A B A B C Ax B Cz D = (3) A A B < l < A B C A B C Ax B Cz D = r n =,, x,, z =,, D () { } l = = z D x D x z Dx D = l D x l z D = l l l l ( l < ) - 3 -

37 Minkowski r D x l z D = l l l l () n = {,,} ( x,, z ) = (,,) D x l z D = l = l ( l > ) l l l l = x z D D x z D D = D x l z D = l l l l ( l < ) l l l l D x l z D = ( l > ) D x l z D = l l l l - 3 -

38 Minkowski r (3) n = {,,} ( x,, z ) = (,,) l = = x D z D x z Dz D = l D z x l D = l l l l l l l l D z x l D = ( l < ) ( l > ) D z x l D = l l l l 5. P( xz,, ) L : x x z z = = l m n ( P x,, z ) κ

39 Minkowski uuur L PP zz zz xx xx m n n l l m n l m ( z z ) ( x x ) ( ) = κ κ κ l m κ ( l m n ) z lmx nlxz mnz ( ) ( ) κ n( mz n) l( l mx) ( l m n ) κ ( ) ( ) κ ( mz n ) ( lz nx ) ( l mx ) m n l m n x l n l m n n lz nx m l mx x l m n x m mz n l lz nx z n l m z ( n l m )( x z ) κ = (5..) uuur L PP zz zz xx xx m n n l l m n l m ( x x ) ( ) ( z z ) = κ κ κ κ ( ) ( ) κ n m l m n x n l l m n l m l m n z lmx nlxz mnz nlznx ml mx x n l m x

40 Minkowski ( ) ( ) n mz n l l mx n l m κ ( ) ( ) m mz n l lz nx z n l m κ z ( mz n ) ( lz nx ) ( l mx ) ( n l m )( x z ) κ = (5..) uuur L PP x x zz zz xx l m m n n l l m n ( z z ) ( x x ) ( ) = κ κ m n κ l m n x l n l m n (5..) l m κ l m n z lmx nlxz mnz ( ) ( ) n lz nx m l mx x n l m κ x ( ) ( ) n mz n l l mx l m n κ ( ) ( ) m mz n l lz nx z n l m κ z ( mz n ) ( lz nx ) ( l mx ) ( n l m )( x z ) κ =

41 Minkowski zz zz xx xx m n n l l m l m n ( z z ) ( x x ) ( ) = κ κ κ n m κ l m n x n l l m n l m l m n z lmx nlxz mnz ( ) ( ) ( ) ( ) nlz nx ml mx x n l m κ x nmz n ll mx n l m κ ( ) ( ) m mz n l lz nx z n l m κ z ( mz n ) ( lz nx ) ( l mx ) ( n l m )( x z ) κ = (5..) uuur L PP x x zz zz xx l m m n n l l m n ( x x ) ( ) ( z z ) = κ κ κ κ ( ) ( ) κ n m l m n x n l l m n l m l m n z lmx nlxz mnz nlznx ml mx x nl m x

42 Minkowski (5..) ( ) ( ) ( ) ( ) m mz n l lz nx z n l m κ z n mz n l l mx n l m κ ( mz n ) ( lz nx ) ( l ) mx ( n l m )( x z ) κ = zz zz xx xx m n n l l m l m n ( x x ) ( ) ( z z ) = κ κ m n κ l m n x l n l m n (5..) l m κ l m n z lmx nlxz mnz ( ) ( ) nlz nx ml mx x n l m κ x ( ) ( ) n mz n l l mx n l m κ ( ) ( ) m mz n l lz nx z n l m κ z ( mz n ) ( lz nx ) ( l mx ) ( n l m )( x z ) κ = 5.. (5..) (5..)

43 Minkowski. (5..) (5..) Ω lm nl lm mn Ω nl mn Ω3 Ω = m n κ l m n Ω = l n κ l m n Ω 3 = l m κ l m n (4..), Φ Φ Φ3 (5..3) Φ = m n κ l m n Φ = ( κ ) κ m n κ ( l m n ) l m n l m n n κ Φ = κ ( κ )( l m n ) ( l m n ) n 3 (4..)

44 Minkowski lm nl m n κ ( l m n ) κ ( l m n ) n mn κ ( l m n ) n (5..) ( κ ) κ m n κ ( l m n ) (5..4) l m n l m n n m n κ ( l m n ) x ( κ ) κ ( l m n ) κ ( l m n ) n z D3 = D 3 lmnx,,,,, z, x,, z. (5..) (5..) (5..5) n m κ l m n lm nl lm n l κ ( l m n ) mn nl mn l m κ l m n (4..) Γ Γ Γ3 (5..6) Γ = n m κ l m n Γ = ( κ ) κ n m κ ( l m n ) l m n l m n n

45 Minkowski κ Γ = κ ( κ )( l m n ) ( l m n ) n 3 (4..) lm nl n m κ ( l m n ) κ ( l m n ) n mn κ ( l m n ) n (5..7) (5..) ( κ ) κ n m κ ( l m n ) l m n l m n n n m κ ( l m n ) x ( κ ) κ ( l m n ) κ ( l m n ) n z D4 = D 4 lmnx,,,,, z, x,, z (5..8) (5..5) (5..8). (5..5) A 3, B 3, C 3 m n κ l m n ( κ ) κ m n κ ( l m n ) ( κ ) κ ( l m n ) κ ( l m n ) n l m n l m n n,, ()

46 Minkowski () κ > n l m n Ax B Cz D = () n m n < κ < l m n l m n Ax B Cz D = (3) κ < n m l m n Ax B Cz D = () () κ > n m l m n A 3 x B3 C3z D3 = () κ < n m l m n. (5..8) A 4, B 4, C 4 n m κ l m n Ax B Cz D = ( κ ) κ n m κ ( l m n ) l m n l m n n,, - 4 -

47 Minkowski ( κ ) κ ( l m n ) κ ( l m n ) n () () κ > Ax B Cz D = () m n l m n < κ < Ax B Cz D = (3) κ < m n l m n Ax B Cz D = () () κ > m n l m n Ax B Cz D = () < κ < m n l m n (3) κ < Ax B Cz D =

48 Minkowski Ax B Cz D = r v =,,,,,, x,, z =,, κ () { } ( x z ) = κ = κ z x κ κ = κ κ κ κ x z = z x κ κ = κ κ κ κ ( ) ( κ > ) κ r z x κ κ = κ κ κ κ ( ) () v = {,, } ( x,, z ) = (,,) ( x,, z ) = (,,) ( κ < ) - 4 -

49 Minkowski κ = κ r x κ z κ = κ κ κ κ x = x κ z κ = κ κ κ κ ( ) x κ z κ = κ κ κ κ ( ) (3) v = {,,} ( x,, z ) = (,,) ( x,, z ) = (,,) κ = z x κ κ = ( κ > ) κ κ κ κ κ z = ( κ < )

50 Minkowski r z x κ κ = κ κ κ κ ( ) z x κ κ = ( ) κ κ κ κ (4) v = {,,} ( x,, z ) = (,,) ( x,, z ) = (,,) κ = x κ z κ = κ κ κ κ κ x = ( κ > ) ( κ < ) x κ z κ = κ κ κ κ ( ) ( κ > ) r x κ z κ = ( ) κ κ κ κ (5) v = {,,} ( x,, z ) = (,,) ( x,, z ) = (,,) ( κ < )

51 Minkowski κ = κ r x κ z κ = κ κ κ κ z x = x κ z κ = κ κ κ κ ( ) x κ z κ = ( ) κ κ κ κ (6) v = {,,} (,, ) (,,) x z = ( x,, z ) = (,,) κ = κ x κ z κ = ( κ > ) ( κ < ) κ κ κ κ z =

52 Minkowski κ κ κ κ x κ z κ = ( ) ( κ > ) x κ z κ = ( ) κ κ κ κ ( κ < )

53 .. [M] [M] 3.. [M] [M], M. Do Carmo and M. Dajczer. Rotation hpersurface of constant curvature [J], Transaction of the American Mathematical Societ,983,77(): N. Ekmekci and K. Ilarslan. On Bertrand Curves and Their Characterization[J], Differential Geometr-Dnamical Sstem,, 3() 7. Baram Sahin, Erol Kilic and Rifat Gunes. Null Helices in R 3 [J], Differential Geometr-Dnamical Sstems,,3() 8. Houh c S. Rotation surfaces of finite tpe[j], Algebras, Group and Geometries,99,7: Chen B Y. Total mean curvature and submanifolds of tpe [M], World Sxientific,984. K.Kenmotsu. Surface of revolution with prescribed mean curvature[j], Tohoku Math.J,98,3: Yan S T. Submanifolds with constant mean curvature [J], Amer J Math,974,96: Otsuki T. Minimal hpersurfaces in a Riemannian manifold of constant curvature[j], Amer J Math, 97,9: Chorng Shi Houh. Rotation Surfaces of Finite Tpe[J],99,7: Vraneeanu G. Surfaces de rotation dans 977,: E [J], Rev Diff Roumaine Math Pures Appl, 5. K.Akutagawa and S. Nishakawa, The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski 3-space[J], Tohoku Math.J., 99,4: Barbosa L, do Carom M. On the size of a stable minimal surface in 976,98: E [J], Amer J Math, 7.. Minkowski [D]

54 8.. Minkowski [D] [M] [M]

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Ρ Τ Π Υ 8 ). /0+ 1, 234) ς Ω! Ω! # Ω Ξ %& Π 8 Δ, + 8 ),. Ψ4) (. / 0+ 1, > + 1, / : ( 2 : / < Α : / %& %& Ζ Θ Π Π 4 Π Τ > [ [ Ζ ] ] %& Τ Τ Ζ Ζ Π ! # % & ( ) + (,. /0 +1, 234) % 5 / 0 6/ 7 7 & % 8 9 : / ; 34 : + 3. & < / = : / 0 5 /: = + % >+ ( 4 : 0, 7 : 0,? & % 5. / 0:? : / : 43 : 2 : Α : / 6 3 : ; Β?? : Α 0+ 1,4. Α? + & % ; 4 ( :. Α 6 4 : & %