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T M
M
C M T M C
M
S
2
1
4
T M
M C M
T M C M
S
2
1
4
G
T M (M, g)
g
T M
M T M
g
G g
T M
(2, 0) O M × M O M
M T M
G
CG
g
G
T M (M, g)
T M
T M M
S
2
R
3
T S
2
T
1
S
2
g
s
1
4
n
M
x
α
: U
α
⊂ R
n
→ M U
α
R
n
M
α
x
α
(U
α
) = M
α, β x
α
(U
α
) ∩ x
β
(U
β
) = W = ∅
x
−1
α
(W ) x
−1
β
(W )
R
n
x
−1
β
◦ x
−1
α
{(U
α
, x
α
)}
(U
α
, x
α
) p ∈ x
α
(U
α
) M p
x
α
(U
α
) p
{(U
α
, x
α
)} 1 2
M
M
n
n M
M
n
1
M
m
2
ϕ : M
1
→ M
2
p ∈ M
1
y : V ⊂ R
m
→ M
2
ϕ(p) x : U ⊂ R
n
→ M
1
p ϕ(x(U)) ⊂ y(V )
y
−1
◦ ϕ ◦ x :⊂ R
n
→ R
m
x
−1
(p) ϕ M
1
p ϕ(p)
M
α : (−, ) → M M α(0) =
p ∈ M
D M p
α t = 0 α
(0) : D → R
α
(0)f =
d
dt
(f ◦ α)
t= 0
f ∈ D.
p t = 0
α : (−, ) → M α(0) = p M p
T
p
M
x : U → M p = x(0)
f α
f ◦ x(q) = f(x
1
, . . . , x
n
), q = (x
1
, . . . , x
n
) ∈ U,
x
−1
◦ α(t) = (x
1
(t), . . . , x
n
(t)),
f α
α
(0)f =
d
dt
(f ◦ α)
t= 0
=
d
dt
f(x
1
(t), . . . , x
n
(t))
t= 0
=
n
i= 1
dx
i
dt
(0)
∂f
∂x
i
0
=
i
dx
i
dt
(0)
∂
∂x
i
0
f,
α
(0)
α
(0) =
n
i= 1
dx
i
dt
(0)
∂
∂x
i
0
∂
∂x
i
0
x
i
→ x(0, . . . , 0, x
i
, 0, . . . , 0)
∂
∂x
1
0
, . . . ,
∂
∂x
n
0
T
p
M
x T
p
M M p
M
n
1
M
m
2
ϕ : M
1
→
M
2
p ∈ M
1
v ∈ T
p
M
1
α : (−, ) → M
1
α(0) = p α
(0) = v
β = ϕ ◦ α dϕ
p
: T
p
M
1
→ T
ϕ(p)
M
2
dϕ
p
(v) = β
(0)
α
dϕ
p
ϕ p
M
1
M
2
ϕ : M
1
→ M
2
ϕ
−1
ϕ
p ∈ M U p V ϕ(p) ϕ : U → V
M
m
1
M
n
2
ϕ : M
1
→ M
2
dϕ
p
: T
p
M
1
→
T
ϕ(p)
M
2
p ∈ M
1
M
m
1
M
n
2
ϕ : M
1
→ M
2
ϕ ϕ(M
1
) ⊂ M
2
ϕ(M
1
)
M
2
ϕ M
1
⊂ M
2
M
1
M
2
X M
p ∈ M X(p) ∈ T
p
M
x : U ⊂ R
n
→ M X
X(p) =
n
i= 1
a
i
(p)
∂
∂x
i
,
a
i
: U → R U
∂
∂x
i
x X
a
i
f : M → R fX
(fX)(p) := f (p)X(p),
X(f )
X(f )(p) := X(p)(f),
X(f ) f X
X, Y M f : M → R
[X, Y ]
[X, Y ](f ) := X(Y (f)) − Y (X(f)),
Co lchete de Lie p ∈ M
[X, Y ](p)(f ) = X(p)(Y (f)) − Y (p)(X(f)).
X, Y, Z M a, b
f, g
[X, Y ] = −[Y, X],
[aX + bY, Z] = a[X, Z] + b[Y, Z],
[[X, Y ], Z] + [[Y, Z], X] + [[Z, X], Y ] = 0,
[fX, gY ] = fg[X, Y ] + fX(g)Y − gY (f)X.
c : I → M
t ∈ I V (t) ∈ T
c(t)
M
V f M t → V (t)f
I
M g p ∈ M
g
p
= ,
p
T
p
M
V W U p
M
p ∈ M → g
p
(V
p
, W
p
)
U
M V =
i
v
i
∂
∂x
i
W =
j
w
j
∂
∂x
j
g
p
(V
p
, W
p
) =
i,j
g
ij
(p)v
i
w
j
g
ij
(p) = g
p
(
∂
∂x
i
(p),
∂
∂x
j
(p))
(U, x)
g =
i,j
g
ij
dx
i
⊗ dx
j
g =
i,j
g
ij
dx
i
dx
j
V ariedade Riem anniana
ϕ : M
n
→ N
n+k
N
ϕ M
u, v
p
= dϕ
p
(u), dϕ
p
(v)
ϕ(p)
, u, v ∈ T
p
M
M ϕ ϕ
M N
f : M → N iso m etria
df
p
(u), df
p
(v)
f(p)
= u, v
p
, ∀p ∈ M, u, v ∈ T
p
M.
M
X(M) C
∞
M
D(M) C
∞
M
∇ M
∇ : X(M) × X(M) → X(M)
∇
fX+g Y
Z = f∇
X
Z + g∇
Y
Z,
∇
X
(Y + Z) = ∇
X
Y + ∇
X
Z,
∇
X
(fY ) = f ∇
X
Y + X(f)Y.
M
∇
V c : I → M
c c
D
dt
(V + W ) =
DV
dt
+
DW
dt
,
D
dt
(fV ) =
df
dt
V + f
DV
dt
,
V Y ∈ X(M )
DV
dt
= ∇
d c
d t
Y.
x : U ⊂ R
n
→ M c(I)∩x(U) = ∅
x
−1
◦ c(t) = (x
1
(t), . . . , x
n
(t)) c(t) t ∈ I
X
i
=
∂
∂x
i
V
V =
j
v
j
X
j
,
v
j
= v
j
(t) X
j
= X
j
(c(t))
∇
X
i
X
j
=
k
Γ
k
ij
X
k
,
DV
dt
=
k
dv
k
dt
+
ij
Γ
k
ij
v
j
dx
i
dt
X
k
.
M
∇ V c : I → M
paralelo
DV
dt
= 0
∀t ∈ I
V c : I → M
n v
k
(t)
dv
k
dt
+
i,j
Γ
k
ij
v
j
dx
i
dt
= 0 k = 1, . . . , n
M
c : I → M M v
0
∈ T
c(t
0
)
M t
0
∈ I
V c
V (t
0
) = v
0
M
∇ , ∇
, c
P , P
c P , P
= cte
M ∇
M V W
c : I → M
d
dt
V, W =
DV
dt
, W + V,
DW
dt
, t ∈ I.
M
XY, Z = ∇
X
Y, Z + Y, ∇
X
Z.
∇
∇
X
Y − ∇
Y
X = [X, Y ], ∀X, Y ∈ X(M).
(U, x) ∇
∇
X
i
X
j
− ∇
X
j
X
i
= [X
i
, X
j
] = 0,
X
i
=
∂
∂x
i
M
∇ M
∇
∇
M
Z, ∇
Y
X =
1
2
XY, Z + Y Z, X − ZX, Y −
−[X, Z], Y [Y, Z], X − [X, Y ], Z
.
X
k
, ∇
X
i
X
j
=
1
2
{X
j
X
i
, X
k
+ X
i
X
k
, X
j
− X
k
X
i
, X
j
}
=
1
2
∂g
ik
∂x
j
+
∂g
kj
∂x
i
−
∂g
ij
∂x
k
,
X
k
, ∇
X
i
X
j
= X
k
,
r
Γ
r
ij
X
r
=
r
Γ
r
ij
X
k
, X
r
=
r
Γ
r
ij
g
rk
,
r
Γ
r
ij
g
rk
=
1
2
∂g
ik
∂x
j
+
∂g
kj
∂x
i
−
∂g
ij
∂x
k
.
(g
ij
) (g
jk
)
Γ
s
ij
=
1
2
k
g
sk
∂g
ik
∂x
j
+
∂g
kj
∂x
i
−
∂g
ij
∂x
k
.
[ij, k] =
1
2
∂g
ik
∂x
j
+
∂g
kj
∂x
i
−
∂g
ij
∂x
k
,
Γ
s
ij
=
k
g
sk
[ij, k]
[ij, l] =
s
g
ls
Γ
s
ij
[ij, k] Γ
k
ij
1
e r
2
do
∂g
ij
∂x
k
= [ik, j] + [jk, i].
γ : I → M
t
0
∈ I
D
dt
(
dγ
dt
) = 0
t
0
γ t ∀t ∈ I
γ
γ : I → M
d
dt
dγ
dt
,
dγ
dt
=
D
dt
dγ
dt
,
dγ
dt
+
dγ
dt
,
D
dt
dγ
dt
= 2
D
dt
dγ
dt
,
dγ
dt
= 0,
dγ
dt
(U, x) γ(t
0
)
U
γ(t) = (x
1
(t), . . . , x
n
(t))
dγ
dt
=
k
dx
k
dt
∂
∂x
k
,
γ
0 =
D
dt
dγ
dt
=
k
d
2
x
k
dt
2
+
i,j
Γ
k
ij
dx
i
dt
dx
j
dt
∂
∂x
k
.
2
da
d
2
x
k
dt
2
+
i,j
Γ
k
ij
dx
i
dt
dx
j
dt
= 0, k = 1, . . . , n.
R M
X, Y ∈ X(M)
R(X, Y ) : X(M) → X(M )
R(X, Y )Z = ∇
Y
∇
X
Z − ∇
X
∇
Y
Z + ∇
[X,Y ]
Z, Z ∈ X(M ),
∇ M
R
X(M) × X(M)
R(f X
1
+ gX
2
, Y
1
) = fR(X
1
, Y
1
) + gR(X
2
, Y
1
),
R(X
1
, fY
1
+ gY
2
) = fR(X
1
, Y
1
) + gR(X
1
, Y
2
),
f, g ∈ D(M), X
1
, X
2
, Y
1
, Y
2
∈ X(M).
X, Y ∈ X(M) R(X, Y ) : X(M) →
X(M)
R(X, Y )(fZ + gW ) = f R(X, Y )Z + gR(X, Y )W,
f, g ∈ D(M) Z, W ∈ X(M ).
R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0,
R(X, Y )Z = − R(Y, X)Z,
R(X, Y )Z, W = −R(X, Y )W, Z,
R(X, Y )Z, W = R(Z, W )X, Y .
(U, x)
p
X
i
=
∂
∂x
i
,
R(X
i
, X
j
)X
k
=
s
R
s
ijk
X
s
,
R
s
ijk
R(X
i
, X
j
)X
k
= ∇
X
j
∇
X
i
X
k
− ∇
X
i
∇
X
j
X
k
+ ∇
[X
i
,X
j
]
X
k
= ∇
X
j
(
l
Γ
l
ik
X
l
) − ∇
X
i
(
l
Γ
l
jk
X
l
)
=
l
∂Γ
l
ik
∂x
j
X
l
+
l
Γ
l
ik
∇
X
j
X
l
−
l
∂Γ
l
jk
∂x
i
X
l
−
l
Γ
l
jk
∇
X
i
X
l
=
l
∂Γ
l
ik
∂x
j
X
l
+
ls
Γ
l
ik
Γ
s
jl
X
s
−
l
∂Γ
l
jk
∂x
i
X
l
−
ls
Γ
l
jk
Γ
s
il
X
s
=
s
l
Γ
l
ik
Γ
s
jl
−
l
Γ
l
jk
Γ
s
il
+
∂Γ
s
ik
∂x
j
−
∂Γ
s
jk
∂x
i
X
s
.
R
s
ijk
=
r
Γ
r
ik
Γ
s
jr
−
r
Γ
r
jk
Γ
s
ir
+
∂Γ
s
ik
∂x
j
−
∂Γ
s
jk
∂x
i
;
R(X
i
, X
j
)X
k
, X
l
=
s
R
s
ijk
X
s
, X
l
=
s
R
s
ijk
g
sl
= R
ijkl
,
R
ijkl
=
s
g
sl
R
s
ijk
.
R
ijkl
=
r
[il, r]Γ
r
jk
−
r
[jl, r]Γ
r
ik
+
∂[ik, l]
∂x
j
−
∂[jk, l]
∂x
i
.
σ ⊂ T
p
M
T
p
M X, Y ∈ σ
K(X, Y ) =
R(X, Y )X, Y
X, XY, Y − X, Y
2
X, Y ∈ σ
p ∈ M
σ ⊂ T
p
M K(X, Y ) = K(σ) {X, Y }
σ σ p
M p M
{e
1
, . . . , e
n
} n = dim M T
p
M R
ijkl
=
R(e
i
, e
j
)e
k
, e
l
i, j, k, l = 1, . . . , n K(p, σ) = K
0
R
ijij
=
−R
ijji
= K
0
i = j R
ijkl
= 0
K(p, σ) = K
0
p ∈ M σ ⊂ T
p
M
M K
0
T M = {(p, v); p ∈ M, v ∈ T
p
M}
M
n
T M
T M M
T M
2n
M
,
{U
α
, x
α
} M (x
1
α
, . . . , x
n
α
)
U
α
{
∂
∂x
1
α
, . . . ,
∂
∂x
n
α
}
x
α
(U
α
)
α
y
α
: U
α
× R
n
→ T M
y
α
(x
1
α
, . . . , x
n
α
, u
1
, . . . , u
n
) =
x
α
(x
1
α
, . . . , x
n
α
),
n
i= 1
u
i
∂
∂x
i
α
,
(u
1
, . . . , u
n
) ∈ R
n
.
{(U
α
× R
n
, y
α
)}
T M
α
x
α
(U
α
) = M,
(dx
α
)
q
(R
n
) = T
x
α
(q)
M, q ∈ U
α
,
α
y
α
(U
α
× R
n
) = T M.
(p, v) ∈ y
α
(U
α
× R
n
) ∩ y
β
(U
β
× R
n
)
q
α
∈ U
α
, q
β
∈ U
β
, v
α
, v
β
∈ R
n
y
−1
β
◦ y
α
(q
α
, v
α
) = y
−1
β
(x
α
(q
α
), dx
α
(v
α
))
= (x
−1
β
◦ x
α
(q
α
), d(x
−1
β
◦ x
α
)(v
α
)).
x
−1
β
◦ x
α
d(x
−1
β
◦ x
α
)
T M
M (p, v) ∈ T M
V, W ∈ T
(p,v)
(T M)
T M
ϕ : t → (α(t), v(t)), ψ : s → (β(s), w(s)),
α(0) = β(0) = p v(0) = w(0) = v V = ϕ
(0), W = ψ
(0)
T M
V, W
(p,v)
=
dα
dt
(0),
dβ
ds
(0)
p
+
Dv
dt
(0),
Dw
ds
(0)
p
.
y : U × R
n
→ T M
(p, v) ϕ ψ
y
−1
◦ ϕ(t) = (x
1
(t), . . . , x
n
(t), v
1
(t), . . . , v
n
(t)),
y
−1
◦ ψ(s) = (¯x
1
(s), . . . , ¯x
n
(s), w
1
(s), . . . , w
n
(s)),
dα
dt
, v
i
,
Dv
i
dt
,
dβ
ds
, w
i
,
Dw
i
ds
t = 0 s = 0
dα
dt
=
j
dx
j
dt
∂
∂x
j
,
Dv
dt
=
j
dv
j
dt
+
l,m
Γ
j
lm
v
m
dx
l
dt
∂
∂x
j
.
dβ
ds
=
k
d¯x
k
ds
∂
∂x
k
,
Dw
ds
=
k
dw
k
ds
+
ri
Γ
k
ri
w
i
d¯x
r
ds
∂
∂x
k
.
∂
∂x
i
,
∂
∂x
j
=
g
ij
V, W
(p,v)
=
g
jk
+ g
lm
Γ
l
jr
Γ
m
ki
v
r
w
i
dx
j
dt
d¯x
k
ds
+ g
jl
Γ
l
ki
w
i
dv
j
dt
d¯x
k
ds
+
+g
lk
Γ
l
js
v
s
dx
j
dt
dw
k
ds
+ g
jk
dv
j
dt
dw
k
ds
.
v(0) = w(0) v
i
(0) = w
i
(0) i = 1, . . . , n
v
j
= x
n+j
w
j
= ¯x
n+j
j = 1 . . . , n
V, W
(p,v)
=
g
jk
+ g
lm
Γ
l
jr
Γ
m
ki
v
r
v
i
dx
j
dt
d¯x
k
ds
+ g
jl
Γ
l
ki
v
i
dx
n+j
dt
d¯x
k
ds
+
+g
lk
Γ
l
js
v
s
dx
j
dt
d¯x
n+k
ds
+ g
jk
dx
n+j
dt
d¯x
n+k
ds
.
G
IJ
G
jk
= g
jk
+ g
ml
Γ
m
rj
Γ
l
sk
v
r
v
s
G
j n+k
= g
mk
Γ
m
lj
v
l
G
n+j k
= g
jm
Γ
m
lk
v
l
G
n+j n+k
= g
jk
[jk, h] = g
sh
Γ
s
jk
G
jk
= g
jk
+ g
ml
Γ
m
rj
Γ
l
sk
v
r
v
s
G
j n+k
= [lj, k]v
l
G
n+j k
= [lk, j]v
l
G
n+j n+k
= g
jk
T M
G
IJ
G
IJ
G
JK
= δ
K
I
,
G
jk
= g
jk
G
j n+k
= −Γ
k
lm
g
jm
v
l
G
n+j k
= −Γ
j
lm
g
km
v
l
G
n+j n+k
= g
jk
+ g
ml
Γ
j
rm
Γ
k
sl
v
r
v
s
i, j, k, . . . 1 n I, J, K, . . . 1 2n
T M
M π : T M → M π(p, v) = p
p ∈ M v ∈ T
p
M
G
IJ
v =
i
v
i
∂
∂x
i
c : I → M
Dv
i
dt
=
dv
i
dt
+ Γ
i
jk
v
j
dx
k
dt
,
D
2
v
i
dt
2
=
d
2
v
i
dt
2
+
d
dt
Γ
i
jk
v
j
dx
k
dt
+Γ
i
lj
Dv
l
dt
dx
j
dt
.
v
c
[IJ , K] Γ
I
JK
1
ra
2
da
[J K, H ] =
1
2
∂G
JH
∂x
K
+
∂G
HK
∂x
J
−
∂G
JK
∂x
H
,
Γ
I
JK
= G
IH
[J K, H ].
1
ra
[n + j n + k, h] =
1
2
∂G
n+j h
∂x
n+k
+
∂G
h n+k
∂x
n+j
−
∂G
n+j n+k
∂x
h
=
1
2
∂
∂v
k
([lh, j]v
l
) +
∂
∂v
j
([lh, k]v
l
) −
∂g
jk
∂x
h
=
1
2
[lh, j]δ
l
k
+ [lh, k]δ
l
j
−
∂g
jk
∂x
h
=
1
2
[kh, j] + [jh, k] −
∂g
jk
∂x
h
= 0.
[n + j n + k, n + h] =
1
2
∂G
n+j n+h
∂x
n+k
+
∂G
n+h n+k
∂x
n+j
−
∂G
n+j n+k
∂x
n+h
=
1
2
∂g
jh
∂v
k
+
∂g
hk
∂v
j
−
∂g
jk
∂v
h
= 0.
[n + j k, n + h] =
1
2
∂G
n+j n+h
∂x
k
+
∂G
n+h k
∂x
n+j
−
∂G
n+j k
∂x
n+h
=
1
2
∂g
jh
∂x
k
+
∂
∂v
j
([lk, h]v
l
) −
∂
∂v
h
([lk, j]v
l
)
=
1
2
[jk, h] + [hk, j] + [jk, h] − [hk, j]
= [jk, h].
[jk, h] =
1
2
∂G
jh
∂x
k
+
∂G
hk
∂x
j
−
∂G
jk
∂x
h
=
1
2
∂
∂x
k
(g
jh
+ g
ml
Γ
m
rj
Γ
l
sh
v
r
v
s
) +
∂
∂x
j
(g
hk
+ g
ml
Γ
m
rh
Γ
l
sk
v
r
v
s
) −
−
∂
∂x
h
(g
jk
+ g
ml
Γ
m
rj
Γ
l
sk
v
r
v
s
)
= [jk, h] +
1
2
∂
∂x
k
(g
ml
Γ
m
rj
Γ
l
sh
) +
∂
∂x
j
(g
ml
Γ
m
rh
Γ
l
sk
) −
∂
∂x
h
(g
ml
Γ
m
rj
Γ
l
sk
)
v
r
v
s
.
[jk, n + h] =
1
2
∂G
j n+h
∂x
k
+
∂G
n+h k
∂x
j
−
∂G
jk
∂x
n+h
=
1
2
∂
∂x
k
([lj, h]v
l
) +
∂
∂x
j
([lk, h]v
l
) −
∂
∂v
h
(g
jk
+ g
ml
Γ
m
rj
Γ
l
sk
v
r
v
s
)
=
1
2
∂
∂x
k
[lj, h]v
l
+
∂
∂x
j
[lk, h]v
l
− g
ml
Γ
m
rj
Γ
l
sk
∂(v
r
v
s
)
∂v
h
=
1
2
∂
∂x
k
[lj, h]v
l
+
∂
∂x
j
[lk, h]v
l
− g
ml
Γ
m
rj
Γ
l
sk
δ
r
h
v
s
− g
ml
Γ
m
rj
Γ
l
sk
δ
s
h
v
r
=
1
2
∂
∂x
k
[lj, h]v
l
+
∂
∂x
j
[lk, h]v
l
− g
ml
Γ
m
hj
Γ
l
sk
v
s
− g
ml
Γ
m
rj
Γ
l
hk
v
r
=
1
2
∂
∂x
k
[lj, h]v
l
+
∂
∂x
j
[lk, h]v
l
− g
ms
Γ
m
hj
Γ
s
lk
v
l
− g
mr
Γ
m
lj
Γ
r
hk
v
l
=
1
2
∂
∂x
k
[lj, h] +
∂
∂x
j
[lk, h] − g
ms
Γ
m
hj
Γ
s
lk
− g
mr
Γ
m
lj
Γ
r
hk
v
l
.
[n + j k, h] =
1
2
∂G
n+j h
∂x
k
+
∂G
hk
∂x
n+j
−
∂G
n+j k
∂x
h
=
1
2
∂
∂x
k
([sh, j]v
s
) −
∂
∂x
h
([sk, j]v
s
) +
∂
∂v
j
(g
hk
+ g
lm
Γ
l
rh
Γ
m
sk
v
r
v
s
)
=
1
2
∂
∂x
k
(g
jr
Γ
r
sh
) −
∂
∂x
h
(g
jr
Γ
r
sk
) + g
lm
Γ
l
jh
Γ
m
sk
+ g
lm
Γ
l
sh
Γ
m
jk
v
s
=
1
2
g
ir
Γ
i
jk
Γ
r
sh
+ g
ij
Γ
i
rk
Γ
r
sh
+ g
jr
∂
∂x
k
Γ
r
sh
− g
ir
Γ
i
jh
Γ
r
sk
− g
ij
Γ
i
rh
Γ
r
sk
−
−g
jr
∂
∂x
h
Γ
r
sk
+ g
lm
Γ
l
jh
Γ
m
sk
+ g
lm
Γ
l
sh
Γ
m
jk
v
s
=
1
2
g
jr
∂
∂x
k
Γ
r
hs
−
∂
∂x
h
Γ
r
ks
+ Γ
l
hs
Γ
r
kl
− Γ
l
ks
Γ
r
hl
+2g
lm
Γ
l
hs
Γ
m
jk
v
s
=
1
2
(g
jr
R
r
hks
+ 2g
lm
Γ
l
hs
Γ
m
jk
)v
s
=
1
2
(R
hksj
+ 2g
lm
Γ
l
hs
Γ
m
jk
)v
s
=
1
2
(R
jskh
+ 2g
lm
Γ
l
hs
Γ
m
jk
)v
s
.
[n + j n + k, H ] = 0,
[n + j k, h] =
1
2
(R
jskh
+ 2g
lm
Γ
l
hs
Γ
m
jk
)v
s
,
[n + j k, n + h] = [jk, h],
[jk, h] = [jk, h] +
1
2
∂
∂x
k
(g
lm
Γ
l
rj
Γ
m
sh
) +
+
∂
∂x
j
(g
lm
Γ
l
rh
Γ
m
sk
) −
∂
∂x
h
(g
lm
Γ
l
rj
Γ
m
sk
)
v
r
v
s
,
[jk, n + h] =
1
2
∂
∂x
k
[sj, h] +
∂
∂x
j
[sk, h] −
−g
lm
Γ
l
hj
Γ
m
sk
− g
lm
Γ
l
sj
Γ
m
hk
v
s
.
2
da
Γ
I
n+j n+k
= G
IH
[n + j n + k, H ]
= 0.
Γ
i
n+j k
= G
ih
[n + j k, h] + G
i n+h
[n + j k, n + h]
=
1
2
g
ih
R
jskh
+ 2g
lm
Γ
l
sh
Γ
m
jk
v
s
− Γ
h
sl
g
il
v
s
[jk, h]
=
1
2
g
ih
R
jskh
v
s
+ g
ih
g
lm
Γ
l
sh
Γ
m
jk
v
s
− g
il
g
hm
Γ
h
sl
Γ
m
jk
v
s
=
1
2
R
i
jsk
v
s
.
Γ
n+i
n+j k
= G
n+i h
[n + j k, h] + G
n+i n+h
[n + j k, n + h]
= −
1
2
Γ
i
rp
g
hp
v
r
R
jskh
+ 2g
lm
Γ
l
sh
Γ
m
jk
v
s
+
+
g
ih
+ g
pq
Γ
i
rp
Γ
h
sq
v
r
v
s
[jk, h]
= −
1
2
Γ
i
rp
R
p
jsk
v
r
− g
lm
g
hp
Γ
i
rp
Γ
l
sh
Γ
m
jk
v
r
v
s
+
+Γ
i
jk
+ g
pq
g
mh
Γ
i
rp
Γ
h
sq
Γ
m
jk
v
r
v
s
= Γ
i
jk
−
1
2
Γ
i
rp
R
p
jsk
v
r
v
s
.
Γ
i
jk
= G
ih
[jk, h] + G
i n+l
[jk, n + l]
= g
ih
[jk, h] +
1
2
∂
∂x
k
(g
lm
Γ
l
rj
Γ
m
sh
) +
∂
∂x
j
(g
lm
Γ
l
sh
Γ
m
rk
) −
−
∂
∂x
h
(g
lm
Γ
l
rj
Γ
m
sk
)
v
r
v
s
−
1
2
Γ
l
sh
g
ih
v
s
∂
∂x
k
[rj, l] +
∂
∂x
j
[rk, l] −
−g
pm
Γ
p
lj
Γ
m
rk
− g
pm
Γ
p
rj
Γ
m
lk
v
r
= Γ
i
jk
+
1
2
g
ih
Γ
l
sh
∂
∂x
k
[rj, l] + [rj, l]
∂
∂x
k
Γ
l
sh
+ Γ
l
sh
∂
∂x
j
[rk, l] +
+[rk, l]
∂
∂x
j
Γ
l
sh
− [rk, p]Γ
l
js
Γ
p
hl
− [rj, p]Γ
m
ks
Γ
p
hm
− [sk, l]
∂
∂x
h
Γ
l
rj
−
−[rj, m ]
∂
∂x
h
Γ
m
sk
− Γ
l
sh
∂
∂x
k
[rj, l] − Γ
l
sh
∂
∂x
j
[rk, l] + [rk, p]Γ
l
sh
Γ
p
lj
+
+[rj, m ]Γ
l
sh
Γ
m
lk
v
r
v
s
= Γ
i
jk
+
1
2
g
ih
[rj, l]
∂
∂x
k
Γ
l
hs
−
∂
∂x
h
Γ
l
ks
+ Γ
m
hs
Γ
l
km
− Γ
m
ks
Γ
l
hm
+
+[rk, l]
∂
∂x
j
Γ
l
hs
−
∂
∂x
h
Γ
l
js
+ Γ
p
hs
Γ
l
jp
− Γ
p
js
Γ
l
hp
v
r
v
s
= Γ
i
jk
+
1
2
g
ih
[rj, l]R
l
hks
+ [rk, l]R
l
hjs
v
r
v
s
= Γ
i
jk
+
1
2
Γ
h
rj
R
i
hsk
+ Γ
h
rk
R
i
hsj
v
r
v
s
.
Γ
n+i
jk
= G
n+i h
[jk, k] + G
n+i n+l
[jk, n + l]
= −Γ
i
tl
g
hl
v
t
[jk, h] +
1
2
∂
∂x
k
(g
lm
Γ
l
rj
Γ
m
sh
) +
∂
∂x
j
(g
lm
Γ
l
rh
Γ
m
sk
) −
−
∂
∂x
h
(g
lm
Γ
l
rj
Γ
m
sk
)
v
r
v
s
+
1
2
g
il
+ g
pm
Γ
i
rp
Γ
l
tm
v
r
v
t
·
∂
∂x
k
[sj, l] +
+
∂
∂x
j
[sk, l] − g
qα
Γ
q
lj
Γ
α
sk
− g
qα
Γ
q
sj
Γ
α
lk
v
s
=
1
2
R
i
skj
+ R
i
sjk
+ 2
∂
∂x
s
Γ
i
jk
v
s
+
1
2
Γ
i
rl
R
l
shk
Γ
h
tj
+ R
l
shj
Γ
h
tk
v
r
v
s
v
t
.
Γ
I
n+j n+k
= 0,
Γ
i
n+j k
=
1
2
R
i
jks
v
s
,
Γ
n+i
n+j k
= Γ
i
jk
−
1
2
Γ
i
sm
R
m
jkr
v
r
v
s
,
Γ
i
jk
= Γ
i
jk
+
1
2
R
i
hsk
Γ
h
rj
+ R
i
hsj
Γ
h
rk
v
r
v
s
,
Γ
n+i
jk
=
1
2
R
i
skj
+ R
i
sjk
+ 2
∂
∂x
s
Γ
i
jk
v
s
+
+
1
2
Γ
i
rl
R
l
shk
Γ
h
tj
+ R
l
shj
Γ
h
tk
v
r
v
s
v
t
.
d
2
x
I
dt
2
+
Γ
I
JK
dx
J
dt
dx
K
dt
= 0, I = 1, . . . , 2n.
M
I = i
d
2
x
i
dt
2
+
Γ
i
jk
dx
j
dt
dx
k
dt
+ 2
Γ
i
n+j k
dv
j
dt
dx
k
dt
+
Γ
i
n+j n+k
dv
j
dt
dv
k
dt
= 0.
d
2
x
i
dt
2
+
Γ
i
jk
+
1
2
Γ
h
rj
R
i
hsk
+ Γ
h
rk
R
i
hsj
v
r
v
s
dx
j
dt
dx
k
dt
+
+ R
i
jsk
v
s
dv
j
dt
dx
k
dt
= 0.
d
2
x
i
dt
2
+ Γ
i
jk
dx
j
dt
dx
k
dt
+ Γ
h
rj
R
i
hsk
v
r
v
s
dx
j
dt
dx
k
dt
+ R
i
jsk
v
s
dv
j
dt
dx
k
dt
= 0,
d
2
x
i
dt
2
= −Γ
i
jk
dx
j
dt
dx
k
dt
− Γ
h
rj
R
i
hsk
v
r
v
s
dx
j
dt
dx
k
dt
− R
i
hsk
v
s
dv
k
dt
dx
k
dt
= −Γ
i
jk
dx
j
dt
dx
k
dt
− R
i
hsk
v
s
dx
k
dt
dv
h
dt
+ Γ
h
rj
v
r
dx
j
dt
= −Γ
i
jk
dx
j
dt
dx
k
dt
+ R
i
shk
v
s
dx
k
dt
Dv
h
dt
.
I = n + i
D
2
v
i
dt
2
=
d
2
v
i
dt
2
+
d
dt
Γ
i
sk
v
s
dx
k
dt
+ Γ
i
sk
dv
s
dt
dx
k
dt
+ Γ
i
sk
v
s
d
2
x
k
dt
2
+ Γ
i
lj
Dv
l
dt
dx
j
dt
D
2
v
i
dt
2
=
d
2
v
i
dt
2
+
d
dt
Γ
i
sk
v
s
dx
k
dt
+ Γ
i
sk
dv
s
dt
dx
k
dt
+ Γ
i
sk
v
s
R
k
rhp
v
r
dx
p
dt
Dv
h
dt
−
−Γ
k
pj
dx
p
dt
dx
j
dt
+Γ
i
lj
dv
l
dt
+ Γ
l
sk
v
s
dx
k
dt
dx
j
dt
=
d
2
v
i
dt
2
+
∂
∂x
j
Γ
i
sk
v
s
dx
j
dt
dx
k
dt
+ Γ
l
sk
Γ
i
lm
v
s
dx
j
dt
dx
k
dt
−
−Γ
i
sl
Γ
l
jk
v
s
dx
j
dt
dx
k
dt
+ R
l
spk
Γ
i
hl
v
s
v
h
dx
k
dt
Dv
p
dt
+ 2Γ
i
jk
dv
j
dt
dx
k
dt
.
I = n + i
d
2
v
i
dt
2
+
Γ
n+i
jk
dx
j
dt
dx
k
dt
+ 2
Γ
n+i
n+j k
dv
j
dt
dx
k
dt
= 0
d
2
v
i
dt
2
+
1
2
R
i
skj
+ R
i
sjk
+ 2
∂
∂x
s
Γ
i
jk
v
s
+
1
2
Γ
i
lh
R
l
spk
Γ
p
rj
+ +R
l
spj
Γ
p
rk
v
r
·
·v
s
v
h
dx
j
dt
dx
k
dt
+ 2
Γ
i
jk
−
1
2
Γ
i
rl
R
l
jsk
v
r
v
s
dv
j
dt
dx
k
dt
= 0
j, k
d
2
v
i
dt
2
+
R
i
sjk
+
∂
∂x
s
Γ
i
jk
v
s
dx
j
dt
dx
k
dt
+ Γ
i
lh
Γ
p
rj
R
l
spk
v
r
v
s
v
h
dx
j
dt
dx
k
dt
+
+2Γ
i
jk
dv
j
dt
dx
k
dt
+ Γ
i
hl
R
l
spk
v
h
v
s
dv
p
dt
dx
k
dt
= 0
d
2
v
i
dt
2
+
∂
∂x
j
Γ
i
sk
+ Γ
l
sk
Γ
i
jl
− Γ
l
jk
Γ
i
sl
v
s
dx
j
dt
dx
k
dt
+
+R
l
spk
Γ
i
hl
v
s
v
h
dx
k
dt
dv
p
dt
+ Γ
p
rj
v
r
dx
j
dt
+2Γ
i
jk
dv
j
dt
dx
k
dt
= 0
d
2
v
i
dt
2
+
∂
∂x
j
Γ
i
sk
v
s
dx
j
dt
dx
k
dt
+ Γ
l
sk
Γ
i
jl
v
s
dx
j
dt
dx
k
dt
−
−Γ
l
jk
Γ
i
sl
v
s
dx
j
dt
dx
k
dt
+ R
l
spk
Γ
i
hl
v
s
v
h
dx
k
dt
Dv
p
dt
+ 2Γ
i
jk
dv
j
dt
dx
k
dt
= 0
d
2
v
i
dt
2
D
2
v
i
dt
2
= 0
d
2
x
i
dt
2
+ Γ
i
jk
dx
j
dt
dx
k
dt
= R
i
jlk
dx
j
dt
v
l
Dv
k
dt
D
2
v
i
dt
2
= 0
i = 1, . . . , n.
M T M
t → γ(t) M
t →
γ(t),
dγ
dt
(t)
T M
G T M
t →
γ(t),
dγ
dt
(t)
γ M
T M
T M
T M
¯α : I → T M t → (α(t), v(t))
v(t) α = π ◦ ¯α T M
α : I → M, t → α(t) p = α(0)
v ∈ T
p
M α
H
: I → T M
(p, v) π ◦ α
H
= α
α
H
(t) = (α(t), v(t))
v(t) α v(0) = v
α
H
α
α M T M
α M
S
2
R
3
T S
2
S
2
g
s
T S
2
1
4
π
1
: T
1
S
2
→ S
2
π
1
(p, v) = p p ∈ S
2
(r, θ) 0 < r < +∞
0 < θ < 2π T
p
S
2
− L L
θ = 0 ex p
p
: T
p
S
2
→ S
2
V T
p
S
2
ex p
p
(V −V ∩L) = U ⊂ S
2
e
1
=
∂
∂r
,
e
2
=
1
sin r
∂
∂θ
,
U π
−1
(U) ⊂
T
1
S
2
(r, θ, ω) → (exp
p
(r, θ), co s ωe
1
+ sin ωe
2
), 0 < ω < 2π.
∂
∂r
∂
∂θ
(r, θ )
r → ex p (r, θ) S
2
D
∂r
e
1
=
D
∂r
∂
∂r
= 0.
e
2
D
∂r
e
1
= 0,
D
∂r
e
2
= 0.
∂
∂θ
= sin re
2
∂
∂θ
,
∂
∂θ
= sin
2
r
2 sin r co s r =
d
dr
∂
∂θ
,
∂
∂θ
= 2
D
∂r
∂
∂θ
,
∂
∂θ
= 2
D
∂θ
∂
∂r
,
∂
∂θ
= 2 sin r
D
∂θ
e
1
, e
2
.
D
∂θ
e
1
, e
2
= co s r,
D
∂θ
e
1
, e
1
=
1
2
d
dθ
e
1
, e
1
= 0,
D
∂θ
e
1
= co s re
2
.
e
1
, e
2
= 0
D
∂θ
e
2
, e
1
= −e
2
,
D
∂θ
e
1
= −e
2
, co s re
2
= − co s r,
D
∂θ
e
2
, e
2
=
d
dθ
e
2
, e
2
= 0.
D
∂θ
e
2
= − co s re
1
.
(r, θ, ω) ∂r, ∂θ, ∂ω
∂r, ∂r
s
=
∂
∂r
,
∂
∂r
+
D
∂r
(co s ωe
1
+ sin ωe
2
),
D
∂r
(co s ωe
1
+ sin ωe
2
)
= 1 + co s ω
D
∂r
e
1
+ sin ω
D
∂r
e
2
, co s ω
D
∂r
e
1
+ sin ω
D
∂r
e
2
= 1.
∂θ, ∂θ
s
=
∂
∂θ
,
∂
∂θ
+
D
∂θ
(co s ωe
1
+ sin ωe
2
),
D
∂θ
(co s ωe
1
+ sin ωe
2
)
= sin
2
r + c o s ω
D
∂θ
e
1
+ sin ω
D
∂θ
e
2
, co s ω
D
∂θ
e
1
+ sin ω
D
∂θ
e
2
= sin
2
r + c o s ω co s re
2
− sin ω c o s re
1
, co s ω co s re
2
− sin ω c o s re
1
= sin
2
r + c o s
2
ω c o s
2
r + sin
2
ω c o s
2
r
= 1.
∂ω, ∂ω
s
= 0, 0 +
D
∂ω
(co s ωe
1
+ sin ωe
2
),
D
∂ω
(co s ωe
1
+ sin ωe
2
)
= − sin ωe
1
+ co s ωe
2
, − sin ωe
1
+ co s ωe
2
= sin
2
ω + co s
2
ω
= 1.
∂r, ∂θ
s
=
∂
∂r
,
∂
∂θ
+
D
∂r
(co s ωe
1
+ sin ωe
2
),
D
∂θ
(co s ωe
1
+ sin ωe
2
)
= 0 + co s ω
D
∂r
e
1
+ sin ω
D
∂r
e
2
, co s ω
D
∂θ
e
1
+ sin ω
D
∂θ
e
2
= 0.
∂r, ∂ω
s
=
∂
∂r
, 0 +
D
∂r
(co s ωe
1
+ sin ωe
2
),
D
∂ω
(co s ωe
1
+ sin ωe
2
)
= co s ω
D
∂r
e
1
+ sin ω
D
∂r
e
2
, − sin ωe
1
+ co s ωe
2
= 0.
∂θ, ∂ω
s
= 0, 0 +
D
∂θ
(co s ωe
1
+ sin ωe
2
),
D
∂ω
(co s ωe
1
+ sin ωe
2
)
= co s ω co s re
2
− sin ω c o s re
1
, − sin ωe
1
+ co s ωe
2
= sin
2
ω c o s r + co s
2
ω c o s r
= co s r.
g
s
= dr
2
+ dθ
2
+ 2 co s rdθdω + dω
2
,
ω
1
= dr,
ω
2
= co s
r
2
(dθ + dω),
ω
3
= sin
r
2
(dθ − dω),
g
s
= (ω
1
)
2
+ (ω
2
)
2
+ (ω
3
)
2
.
{ω
1
, ω
2
, ω
3
} (g
ij
= δ
ij
)
ω
i
j
g
s
dω
i
=
3
j= 1
ω
j
∧ ω
i
j
, i = 1, 2, 3,
ω
i
j
= −ω
j
i
.
f = co s
r
2
h = sin
r
2
ω
1
= dr,
ω
2
= f (dθ + dω),
ω
3
= h(dθ − dω).
dω
1
= 0.
dω
2
= df ∧ (dθ + dω)
= f
dr ∧ (dθ + dω)
=
f
f
ω
1
∧ ω
2
.
dω
3
= dh ∧ (dθ − dω)
= h
dr ∧ (dθ − dω)
=
h
h
ω
1
∧ ω
3
.
dω
i
ω
1
2
= −ω
2
1
= −
f
f
ω
2
,
ω
1
3
= −ω
3
1
= −
h
h
ω
3
,
ω
i
j
= 0,
.
Ω
i
j
Ω
i
j
= dω
i
j
+
3
k= 1
ω
i
k
∧ ω
k
j
.
dω
2
1
= d(
f
f
ω
2
)
= d(f
(dθ + dω))
= df
∧ (dθ + dω)
= f
dr ∧ (dθ + dω)
=
f
f
ω
1
∧ ω
2
.
dω
3
1
=
h
h
ω
1
∧ ω
3
.
Ω
2
1
= dω
2
1
+
3
k= 1
ω
2
k
∧ ω
k
1
= dω
2
1
+ 0
=
f
f
ω
1
∧ ω
2
.
Ω
3
1
= dω
3
1
+
3
k= 1
ω
3
k
∧ ω
k
1
= dω
3
1
+ 0
=
h
h
ω
1
∧ ω
3
.
Ω
3
2
= dω
3
2
+
3
k= 1
ω
3
k
∧ ω
k
2
= 0 + ω
3
1
∧ ω
1
2
=
f
h
fh
ω
2
∧ ω
3
.
f = co s
r
2
h = sin
r
2
Ω
2
1
= −
1
4
ω
1
∧ ω
2
,
Ω
3
1
= −
1
4
ω
1
∧ ω
3
,
Ω
3
2
= −
1
4
ω
2
∧ ω
3
,
R g
s
{ω
1
, ω
2
, ω
3
}
Ω
ij
= −
1
2
3
k,l= 1
R
ijkl
ω
k
∧ ω
l
= −
1≤k< l≤3
R
ijkl
ω
k
∧ ω
l
,
Ω
ij
= g
jk
Ω
k
i
= Ω
j
i
R
1212
= R
1313
= R
2323
=
1
4
.
T
1
S
2
1
4
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