On an invariant on isometric immersions into spaces of constant sectional curvature
In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension...
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| Дата: | 2009 |
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Institute of Mathematics, NAS of Ukraine
2009
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509165870383104 |
|---|---|
| author | Rivertz, H. J. Ріверц, Х. Дж. |
| author_facet | Rivertz, H. J. Ріверц, Х. Дж. |
| author_sort | Rivertz, H. J. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:45:55Z |
| description | In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups $G$ with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of $G$ is not symmetric, then there are no local isometric immersions of $G$ into $Q_{c^4}$. |
| first_indexed | 2026-03-24T02:36:46Z |
| format | Article |
| fulltext |
UDC 517.9
H. J. Rivertz (Norweg. Univ. Sci. and Technol.)
ON AN INVARIANT ON ISOMETRIC IMMERSIONS
INTO SPACES OF CONSTANT SECTIONAL CURVATURE*
PRO INVARIANT NA IZOMETRYÇNYX ZANURENNQX
U PROSTORY STALO} SEKCIJNO} KRYVYNY
In the present paper we give an invariant on isometric immersions into spaces of constant sectional
curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of
isometric immersions. We will apply this invariant on some examples. Further we will apply it to
codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary left invariant
Riemannian metric into spaces of constant non-positive sectional curvature. We will also consider the
more general class: Three dimensional Lie groups G with non-trivial center and with arbitrary left-
invariant metric. We show that when the metric of G is not symmetric, there are no local isometric
immersions of G into Qc
4 .
Navedeno invariant na izometryçnyx zanurennqx u prostory stalo] sekcijno] kryvyny. Cej inva-
riant [ naslidkom rivnqnnq Haussa ta rivnqnnq Kodacci dlq izometryçnoho zanurennq. Cej inva-
riant vykorystano u dekil\kox prykladax. Joho zastosovano do lokal\nyx izometryçnyx zanu-
ren\ kovymirnosti 1 2-krokovyx nil\potentnyx hrup Li z dovil\nog invariantnog zliva rimano-
vog metrykog u prostir stalo] nedodatno] sekcijno] kryvyny. Rozhlqnuto takoΩ bil\ß zahal\-
nyj klas, a same tryvymirni hrupy Li G iz netryvial\nym centrom ta dovil\nog invariantnog
zliva metrykog. Pokazano, wo u vypadku, koly metryka G nesymetryçna, ne isnu[ lokal\nyx
izometryçnyx zanuren\ G u Qc
4
.
1. Introduction. A special case of a result, due to Otsuki [1] is as follows. If the
sectional curvature of a Riemannian manifold is strictly negative, then any local
isometric immersion of the Riemannian manifold into a Euclidean space is of
codimension greater or equal to one less than the dimension of the manifold. The
sectional curvature of a 2-step nilpotent group is not strictly negative. By studying the
Gauss equation for the curvature tensor of 2-step nilpotent groups with arbitrary left
invariant metric, it has been shown [2] that there exists no isometric immersions of
open subsets of the 2-step nilpotent groups into Euclidean space.
Recently Masal’tsev [3] proved that there are no isometric immersions of any
region of the three dimensional Heisenberg group into any space form of constant
sectional curvature. In the present article we shall prove that no left invariant
nonsymmetric metrics of the three dimensional Lie groups of Bianchi type II and III
are not immersable into a four dimensional space-forms of constant sectional curvature.
2. A new invariant. Let * denote the Hodge star operator, and let R : Λ2
T M →
→ Λ2
T M denote the curvature operator of a Riemannian manifold M. That is 〈 R ( X ∧
∧ Y ), Z ∧ W 〉 = 〈 R ( X, Y ) W, Z 〉.
Theorem 1. Let M be a three dimensional Riemannian manifold and let p be
an arbitrary chosen point in M. Let Ξ is an endomorphism of T Mp which
permuting two of three vectors and fixes the third vector in some chosen basis { e1 , e2 ,
e3 } of T Mp . Let C Ξ : T Mp → Λ 2
T Mp be the map defined by CΞ : Xp �
� (∇ ) ( )( ) ( )Ξ ΞX p pp
R X X∧ . Define the number fM p, by
f R cI I C e eM p cf
t
i i
i
, * ,= ( − ) ( )〈 〉
=
∑ � �∧ Ξ
1
3
,
*
The work was partly supported by NFR Grant #110892/410.
© H. J. RIVERTZ, 2009
1660 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
ON AN INVARIANT ON ISOMETRIC IMMERSIONS INTO SPACES … 1661
where ( − )R cI I cf
t∧ : Λ 2
( Tp M ) → Λ 2
( Tp M ) denotes the transposed cofactor
operator of R cI I− ∧ with respect to the basis { e1 ∧ e2 , e1 ∧ e3 , e2 ∧ e3 }. If a
neighborhood of p in M is isometric immersable into Qc
4 , we have fM p, = 0.
Remark 1. The invariant fM p, in Theorem 1 was found by a method given in my
thesis [4]. The method used in [4] generalized a method by Agaoka [5] which uses
G L ( n )-equivariance of the Gauss equation. A paper that extends the method from my
thesis is in preparation [6].
Corollary 1. Let M be a Riemannian space of dimension 3 with metric 〈 , 〉
and let p be an arbitrary chosen point in M . Let X1, X2 and X3 be arbitrary
vectors in Tp M.
Define the function fM p, by
f X X X R C R R CM p, , ,( ) = −1 2 3 1213
2
21223 1313 1212 212233 1223 1213 11223− R R C +
+ R R C R R C R R1323 1212 11223 1213 1223 21213 1323 121− + 22 21213C +
+ R C R R C R R C1223
2
11213 2323 1212 11213 1313 1223 21− + 2212 –
– R R C R R C R R1323 1213 21212 1323 1223 11212 2323 121− + 33 11212C ,
where
C R X X X Xijklm X j k l mi
= (∇ )( )〈 〉, ,
and
R R X X X X c X X X X c Xijkl i j k l i l j k i= ( ) − 〈 〉〈 〉 + 〈〈 〉, , , , , XX X Xk j l〉〈 〉,
for i, j, k , l, m = 1, 2, 3. If an open neighborhood of p i n M is isometric
immersable into a 4-dimensional space Qc
4 of arbitrary constant sectional
curvature c, we have f X X XM p, , ,( )1 2 3 = 0 for all X1 , X2 , and X3 ∈ Tp M.
Remark 2. In the corollary, we have used a Ξ which interchanges X1 and X2 .
Theorem 2. Let G be an 3-dimensional 2-step nilpotent group G equipped
with an arbitrary left invariant metric. If there is a set X , Y , Z of O . N . left
invariant vector fields such that [ X, Y ] = Z and with ad *( )X Z ≠ ad *( )Y Z or
c ≠
ad *( )X Z
Z4
2
, then there exists no isometric immersions of an open set of G into a
space Qc
4 of sectional curvature c.
We will prove this theorem in Subsection 4.2.
3. On local isometric immersions. Let M denote a Riemannian manifold and let
p be a fixed point in M. Given a local isometric immersion of M into Qc
4 . Let α
be its second fundamental form and let β be the covariant derivative of α. Recall
that by the Codazzi equation, β is symmetric. Define L and B by 〈 L ( X ), Y 〉 =
= 〈 α ( X, Y ), ξ 〉 and 〈 B ( X, Y ), Z 〉 = 〈 β ( X, Y, Z ), ξ 〉, where X , Y, Z ∈ Tp G are
tangent vectors at e and ξ ∈ Np G is a normal vector of the immersion at p.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
1662 H. J. RIVERTZ
Proof of Theorem 1. The Gauss equation for isometric immersions into spaces of
constant sectional curvature c is R ( X ∧ Y ) = – L ( X ) ∧ L ( Y ) + c X ∧ Y (see, e.g., [7].)
The cofactor version of the Gauss equation is ( − )R cI I cf
t∧ = − L Lcf
t
cf
t∧ . The
prolonged Gauss – Codazzi equation is ( ∇T R ) = – L ∧ B ( T, • ) – B ( T, • ) ∧ L (see [8]
for the case of Euclidean ambient space and Corollary 2 in the present article for the
case when the ambient space has constant sectional curvature). Hence, CΞ ( X ) =
= – L ( Ξ ( X ) ) ∧ B ( Ξ ( X ), X ) – B ( Ξ ( X ), Ξ ( X ) ) ∧ L ( X ). Therefore,
( − ) ( ) = ( ) ( ) ( )( )R cI I C X L X L B X Xcf
t
cf
t∧ ∧� �Ξ Ξ Ξdet , +
+ det ,( ) ( ) ( )( )L L B X X Xcf
t � Ξ Ξ ∧ .
Now, since 〈 〉 ( )( ) =* , detX Y Z X Y Z∧ , we have
〈 〉 (( − ) ( ) = ( ) ( )* , det det� � �R cI I C X X L X Lcf
t
cf
t∧ Ξ Ξ BB X X X( ) )( )Ξ , +
+ det det ,( ) ( ) ( )( ( ) )L L B X X X Xcf
t � Ξ Ξ =
= det det det ,( ) ( ) ( ) ( ) ( ) −( ( ) ( ) )L L L X B X X L Xcf
t Ξ Ξ 0 =
= − ( ) ( ) ( ) ( )( ( ( )))det det ,L L X L X B X X3 Ξ Ξ =
= det * , ,( ) − ( ) ( )〈 ( ) ( )〉L L L X X B X X3 � ∧ ∧Ξ Ξ =
= det * , ,( ) ( ) − ( ) ( )〈 ( ) ) ( )〉L R X X c X X B X X3 � Ξ Ξ Ξ∧ ∧ .
We can assume that Ξ permutes e1 and e2 . Therefore,
f L R e e ce e B e eM p, det * , ,= ( ) ( ) − ( )〈 ( ) 〉3
2 1 2 1 2 1∧ ∧ +
+ det * , ,( ) ( ) − ( )〈 ( ) 〉L R e e ce e B e e3
1 2 1 2 1 2∧ ∧ = 0.
The theorem is proved.
4. On 2-step nilpotent groups. 4.1. On the Levi-Civita connection of 2-step
nilpotent groups. We recall some facts about 2-step nilpotent groups. Let � be a Lie
algebra. Define �i , i ≥ 0, recursively by �0 = � and �n = [ �n – 1 , � ] for n ∈ N.
Definition. A Lie algebra is called nilpotent if �n = { 0 } for some integer n. If
�k = { 0 } and �k – 1 ≠ { 0 } then � is called k-step nilpotent. A Lie group G i s
called k-step nilpotent if its Lie algebra is k-step nilpotent.
Let G be a Lie group equipped with an arbitrary left invariant inner product 〈 , 〉. If
X and Y are left invariant vector fields on G then 〈 X, Y 〉 is constant. Therefore,
∇ = [ ] − ( ) − ( )XY X Y X Y Y X
1
2
1
2
1
2
, ad ad* * , (1)
where ad *( )X denotes the adjoint map of ad ( )X .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
ON AN INVARIANT ON ISOMETRIC IMMERSIONS INTO SPACES … 1663
Let G be a 2-step nilpotent Lie group. Let X and Y be orthonormal left invariant
vector fields perpendicular to �1 such that [ X, Y ] ≠ 0. Let Z = [ X, Y ]. We easily
compute ∇ = −∇ =X YY X Z
1
2
, ∇ = ∇ = − ( ) ⊥X ZZ X X Y
1
2
ad * �1 , ∇Y Z =
= ∇ = − ( ) ⊥ZY Y Z
1
2
ad * �1 , and ∇ = ∇ = ∇X Y ZX Y Z = 0. Since Z is independent
of rotations of X and Y we may replace X and Y with the orthonormal eigen vectors
of the symmetric bilinear form
( ) ( ) ( )〈 〉U V U Z V Z, ad , ad* *�
on span { X, Y }. From [2] we have that
〈 〉( ) − ( ) = +R X Y X R X Y X Y Z c, , ,� 3
4
2
,
〈 〉( ) − ( )R X Y X R X Y X Z, , ,� = 0,
〈 〉( ) − ( )R X Y Y R X Y Y Z, , ,� = 0,
(2)
〈 〉( ) − ( ) = − ( ) +R X Z X R X Z X Z X Z c Z, , , ad *� 1
4
2 2
,
〈 〉( ) − ( )R X Z Y R X Z Y Z, , ,� = 0,
〈 〉( ) − ( ) = − ( ) +R Y Z Y R Y Z Y Z Y Z c Z, , , ad *� 1
4
2 2
,
where �R denotes the curvature tensor of a space form of constant sectional curvature
c. We easily compute the covariant derivatives of the curvature tensor by using the
formulas in (2) and the formula (1): The nonzero components are
〈 〉(∇ )( ) = ( )X R X Y X Z X Z, , ad *
1
2
2
,
(3)
〈 〉(∇ )( ) = ( )Y R X Y Y Z Y Z, , ad *
1
2
2
.
We have only considered formulas involving X, Y, and Z.
4.2. Proof of Theorem 2. Let X, Y be orthonormal left invariant vector fields
perpendicular to �1 , and let Z = [ X, Y ]. The invariant fM p, for arbitrary p ∈ G = M
in Corollary 1 together with the formulas in (1) and (2) gives
f X Z Y R R CG p XZXZ YZYZ XXYXZ, , ,( ) = − = 0,
f X Y Z R R C R R CG p XZXZ XYXY YXYYZ YZYZ XYXY XX, , ,( ) = − − YYXZ = 0.
The last pair of equations yields ad ad* *( ) = ( )X Z Y Z and c =
ad *( )X Z
Z
2
2
4
.
The theorem is proved.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
1664 H. J. RIVERTZ
5. An application of the main theorem on Lie groups of nontrivial center. Let
G be a three dimensional Lie group with an arbitrary left invariant metric 〈 , 〉 and with
a nontrivial center of its Lie algebra. Let e1 ∈ � be a left invariant unit vector field on
G contained in the center Z ( � ) of �. From (1) one has ∇e e
1 1 = 0. One can extend
{ e1 } to an orthogonal basis { e1 , e2 , e3 } of left invariant vector fields on G. Let Γ ij
k
be the k-th coefficient of ∇e e
i j . From the compatibility conditions of ∇, we have
Γ Γij
k
j k k j ik
j
i i
= 〈∇ 〉 = − 〈∇ 〉 = −e ee e e e, , . The identities Γ1 j
k = Γ j
k
1 and
Γ Γ Γ11 22
1
33
1k = = = 0 follows from the equation (1) and e1 ∈ Z ( � ). An easy
calculation gives
〈 〉( ) − ( ) = − ( )R R ce e e e e e e1 2 1 1 2 1 2 23
1 2, , ,� Γ ,
〈 〉( ) − ( ) = − ( )R R ce e e e e e e1 3 1 1 3 1 3 23
1 2, , ,� Γ , (4)
〈 〉( ) − ( ) = − ( ) + [R R ce e e e e e e e e2 3 2 2 3 2 3 23
1 2
2, , , ,� Γ 33
2]
and
〈 〉(∇ )( ) = − [ ]e e e e e e e
2 1 2 2 3 23
1
2 3
2
R , , ,Γ ,
(5)
〈 〉(∇ )( ) = − [ ]e e e e e e e
3 1 3 2 3 23
1
2 3
2
R , , ,Γ ,
where we only have displayed the nonzero formulas up to the symmetries of R, �R ,
and ∇R. We now calculate the invariant fG p, in Corollary 1:
f cG p, , , ,( ) = − ( ) [ ]( )e e e e e1 2 3 23
1
23
1 2 2
2 3
2Γ Γ .
Proposition 1. Let G be a three dimensional Lie group with a nontrivial center
of its Lie algebra � and with a left invariant metric 〈 , 〉 such that the center is not
perpendicular to the derived algebra �1 = [ �, � ]. If
c ≠ ( )Γ23
1 2 ,
then there exists no isometric immersions of any region of G into Qc
4 . In
particular Γ23
1 ≠ 0, so there exists no isometric immersions of any region of G
into R4.
Remark 3. For G = Nil3 one has �1 = Z ( � ), so the theorem applies for all left
invariant geometries on Nil3.
Proof of Proposition 1. Since �1 = span { [e2 , e3 ] } and Z ( � ) ⊥ �1
, we have
Γ23
1 ≠ 0 and | [e2 , e3 ] | ≠ 0. Thus, fG p, , ,( )e e e1 2 3 ≠ 0, and by Theorem 1 there are
no isometric immersion of any region of G into Qc
4 .
This is how long we come by using our invariant fG p, . Let c = ( )Γ23
1 2 . By the
same methods as in [3], we show that the following lemma.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
ON AN INVARIANT ON ISOMETRIC IMMERSIONS INTO SPACES … 1665
Lemma. Let G be as in Proposition 1. The second fundamental form of the
isometric immersion of G into Qc
4 is on the form
L = [ ]
−
e e2 3
0 0 0
0
0
, cos sin
sin cos
θ θ
θ θ
,
where θ is a smooth function on G.
Proof. The Gauss equations yield L L L11 12 13= = = 0. The Codazzi equation,
0 = ∇ ( ) − ∇ ( )e ee e e e
2 33 1 2 1L L, , , reduces to − =Γ Γ23
1
22 23
1
33L L . Therefore, L22 =
= − L33 . The third equation in (4) now gives the result.
The lemma is proved.
The covariant derivative of L is
∇ = ( ) − −( )e e
1 1 13
2
23 22
22 23
2
0 0 0
0
0
L L L
L L
θ Γ ,
∇ =
−
− ( ) +(e e
2
0
2
13
2
23 13
2
22
13
2
23 23 2 22L
L L
L L
Γ Γ
Γ Γθ 33
22 2 22
3
13
2
22 22 2 22
3
2
2
) ( )
( )
( ) +
− ( ) +
L
L L
e
e
θ
θ
Γ
Γ Γ LL23 2 22
32( )( ) +e θ Γ
,
∇ =
− −
− − ( ) −(e e
3
0
2
13
2
22 13
2
23
13
2
22 23 3L
L L
L L
Γ Γ
Γ Γθ 333
2
22 3 33
2
13
2
23 22 3 33
2
2
) ( )
(
( ) −
− ( ) −
L
L L
e
e
θ
θ
Γ
Γ Γ22
23 3 33
22) ( )( ) −L e θ Γ
.
The nontrivial Codazzi equations are:
0 = (∇ ) − (∇ ) = − ( ) −( )e e e
1 222 12 1 13
2
23L L Lθ Γ ,
0 = (∇ ) − (∇ ) = ( ) −( )e e e
1 223 13 1 13
2
22L L Lθ Γ ,
0 = (∇ ) − (∇ ) = ( ) −( )e e e
1 332 12 1 13
2
22L L Lθ Γ ,
0 = (∇ ) − (∇ ) = ( ) −( )e e e
1 333 13 1 13
2
23L L Lθ Γ ,
0 = (∇ ) − (∇ ) = ( ) + (( ) + (e e e e
2 332 22 22 2 22
3
23 32L L L Lθ θΓ )) − )2 33
2Γ ,
0 = (∇ ) − (∇ ) = ( ) + (( ) − (e e e e
2 333 23 23 2 22
3
22 32L L L Lθ θΓ )) − )2 33
2Γ .
Therefore, the Codazzi equations are equivalent with
dθ( ) =e1 13
2Γ ,
dθ( ) = −e2 22
32Γ , (6)
dθ( ) =e3 33
22Γ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
1666 H. J. RIVERTZ
Let ω1 , ω2 , ω3 be the dual forms of the vectors e1 , e2 , e3 . Hence, the equations in
(6) can be written on the compact form
dθ ω ω ω= − +Γ Γ Γ13
2
1 22
3
2 33
2
32 2 .
The integrability condition for is
0 = Γ Γ Γ13
2
1 22
3
2 33
2
32 2d d dω ω ω− + . (7)
We calculate dω1 13
2
2 32= − Γ e e∧ , dω2 22
3
2 3= Γ e e∧ , and dω2 33
2
2= − Γ e ∧ e3 .
By substituting these for d kω , k = 1, 2, 3, in (7) we get
0 = − ( ) − ( ) − ( )2 2 223
1 2
22
3 2
33
2 2Γ Γ Γ . (8)
So, condition (8) is necessary for the solutions of the Codazzi equation. This condition
is impossible since we have assumed that Γ23
1 ≠ 0. Therefore:
Theorem 3. Let G be a three dimensional Lie group with a nontrivial center of
its Lie algebra �. Let G haves a left invariant metric 〈 , 〉. If the center of � is
not perpendicular to the derived algebra �1 = [ �, � ], then there are no isometric
immersions of any region of G into Qc
4 .
Remark 4. If �1 ⊥ Z ( � ), then Γ23
1 = 0 and hence, from (5), G is local
symmetric. This case is not considered in this article.
6. The Gauss – Codazzi equation. Let M be an n-dimensional manifold, and let
�N be an m-dimensional manifold. In this appendix we will state first prolongation of
the Gauss and Codazzi equations. The special case for isometric immersions into
Euclidean space is proved by Kaneda [8].
Let 〈 •, • 〉 M and 〈 •, • 〉 be Riemannian metrics on the manifolds M and �N
respectively. Let f be an isometric immersion between the Riemannian manifolds M
and �N with Levi-Civita connections ∇ and �∇ respectively.
Proposition 2 (First prolonged Gauss – Codazzi equation). Let f be an isometric
immersion of M into �N . The following equation is satisfied:
〈 〉 〈 〉(∇ )( ) = (∇ )( )� �
V VR X Y Z W R X Y Z W, , , , –
– 〈 〉 〈 〉(∇ )( ) ( ) − (∇ )( ) ( )⊥ ⊥
V VX W Y Z Y Z X Wα α α α, , , , , , +
+ 〈 〉 〈 〉(∇ )( ) ( ) + (∇ )( ) ( )⊥ ⊥
V VX Z Y W Y W X Zα α α α, , , , , , –
– 〈 〉 〈 〉( ) ( ) + ( ) ( )� �R X Y Z W V R X Y W Z V, , , , , ,α α +
+ 〈 〉 〈 〉( ) ( ) − ( ) ( )� �R Z W Y X V R Z W X Y V, , , , , ,α α ,
where α is the second fundamental form of the immersion.
Proof. The proof is a result of a straight forward calculation.
Let A be the shape operator (see, e.g., [9]).
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
ON AN INVARIANT ON ISOMETRIC IMMERSIONS INTO SPACES … 1667
Corollary 2. Let { … }−ξ ξ1, , m n be a local orthonormal frame of unit normal
vectors, and assume that �N is of constant sectional curvature. We then have
0 = (∇ )( ) + ( ) (∇ ) − ( ) (∇ )∑V V
i
VR X Y A X A Y A Y A
i i i i
, ξ ξ ξ ξ∧ ∧ XX
i
∑ .
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Received 30.01.08,
after revision — 26.09.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 12
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| id | umjimathkievua-article-3129 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:36:46Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a7/6575f47e8cb4d7440869193029b7b3a7.pdf |
| spelling | umjimathkievua-article-31292020-03-18T19:45:55Z On an invariant on isometric immersions into spaces of constant sectional curvature Про інваріант на ізометричних зануреннях у простори сталої секційної кривини Rivertz, H. J. Ріверц, Х. Дж. In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups $G$ with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of $G$ is not symmetric, then there are no local isometric immersions of $G$ into $Q_{c^4}$. Наведено інваріант на ізометричних зануреннях у простори сталої секційної кривини. Цей інваріант є наслідком рівняння Гаусса та рівняння Кодацці для ізометричного занурення. Цей інваріант використано у декількох прикладах. Його застосовано до локальних ізометричних занурень ковимірності 1 2-крокових нільпотентних груп Лі з довільною інваріантною зліва рімановою метрикою у простір сталої недодатної секційної кривини. Розглянуто також більш загальний клас, а саме тривимірні групи Лі $G$ із нетривіальним центром та довільною інваріантною зліва метрикою. Показано, що у випадку, коли метрика $G$ несиметрична, не існує локальних ізометричних занурень $Q_{c^4}$. Institute of Mathematics, NAS of Ukraine 2009-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3129 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 12 (2009); 1660-1704 Український математичний журнал; Том 61 № 12 (2009); 1660-1704 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3129/3006 https://umj.imath.kiev.ua/index.php/umj/article/view/3129/3007 Copyright (c) 2009 Rivertz H. J. |
| spellingShingle | Rivertz, H. J. Ріверц, Х. Дж. On an invariant on isometric immersions into spaces of constant sectional curvature |
| title | On an invariant on isometric immersions into spaces of constant sectional curvature |
| title_alt | Про інваріант на ізометричних зануреннях у простори сталої секційної кривини |
| title_full | On an invariant on isometric immersions into spaces of constant sectional curvature |
| title_fullStr | On an invariant on isometric immersions into spaces of constant sectional curvature |
| title_full_unstemmed | On an invariant on isometric immersions into spaces of constant sectional curvature |
| title_short | On an invariant on isometric immersions into spaces of constant sectional curvature |
| title_sort | on an invariant on isometric immersions into spaces of constant sectional curvature |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3129 |
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