Frame bundle approach to generalized minimal submanifolds

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by  Kamil Niedzialomski

FRAME BUNDLE APPROACH TO GENERALIZED MINIMAL
SUBMANIFOLDS
KAMIL NIEDZIAL OMSKI
Abstract. We extend the notion of r–minimality of a submanifold in arbi-
6
1 trary codimension to u–minimality for a multi–index u ∈ Nq, where q is the
0 codimension. This approach is based on the analysis on the frame bundle of
2 orthonormal frames of the normal bundle to a submanifold and vector bun-
n dles associated with this bundle. The notion of u–minimality comes from the
a variation of σ –symmetric function obtained from the family of shape opera-
u
J
tors corresponding to all possible bases of the normal bundle. We obtain the

9 variation field, which gives alternative definition of u–minimality. Finally, we
1
givesomeexamplesofu–minimalsubmanifoldsforsomechoicesofuandstate

some relations between generalized symmetric functions σ .
] u
G
D
.
h
t
a
m 1. Introduction
[
The notion of minimality of a submanifold has long history. It has been con-

2 sidered by many authors within many contexts. Minimality can be defined by
v
vanishing of the mean curvature – the trace of the second fundamental form. The
8
4 other possible extrinsic conditions we may impose on a submanifold are: total
2 geodesicity, when second fundamental form vanishes, and total umbilicity, when
2
the second fundamental form is proportional to the mean curvature vector.
0
. Whilestudying hypersurfaces, i.e. codimension one submanifolds, we may con-
1
0 sider more invariants, which come from eigenvalues of the second fundamental
6 form. We define the r–th curvature as a r–th symmetric function of the principal
1
curvatures. This has been extensively considered since Reilly’s work [17]. Let us
:
v
briethly recall his approach. Let L be a hypersurface in a Riemannian manifold
i
X
(M,g) with the Levi–Civita connection ∇. Denote by B the second fundamental
r form and by A = AN the shape operator corresponding to a choice of normal
a
vector N,
B(X,Y) = (∇ Y) , g(A(X),Y) = g(B(X,Y),N), X,Y ∈ TL.
X ⊥
The symmetric function S of A is defined by the characteristic polynomial
r
n
χ (t) = det(I +tA) = S tr,
A r
r=0
X
2010 Mathematics Subject Classification. 53C40; 53C42.
Key words and phrases. Submanifold; mean curvatures; first variation; u–minimal subman-
ifold; generalized Newton transformation.
1
2 K.NIEDZIAL OMSKI
where n = dimL. It is a remarkable observation that the variation of S is
r
described by the Newton transformation T = T (A). Namely, let
r r
r
T = (−1)jS Ar j, r = 0,1,...,n.
r j −
j=0
X
Then, we have
d d
(1.1) S (t) = tr A(t)·T ,
r r 1
dt dt −
(cid:18) (cid:19)
where S (t) is a one parameter family of symmetric functions corresponding to a
r
family of operators A(t), which satisfies A(0) = A [17]. With the use of the above
formula Reilly obtained the variation of the integral of S in the case of M being
r
of constant sectional curvature. The Euler–Lagrange equation defines the notion
of r–minimality. Studies on submanifolds satisfying certain conditions involving
r–th mean curvature has been recently very fruitful [1, 2, 7, 8, 13]. Analogous
considerations has been also led in the case of foliations [7, 4].
There has been several attempts to generalize this approach to submanifolds
of arbitrary codimension (see [18, 10, 11]) and to the case of arbitrary foliations
[9, 5]. The problems which appear are the following. First of all, the normal
bundle to a submanifold is not trivial in a sense that the covariant derivative ∇
⊥
does not annihilate unit vector fields. Secondly, there is no canonical choice of the
orthonormal basis in the normal bundle. The second problem has been overcome
(see [18, 14]) by introducing transformations which depend only on the points on
the manifold,
1
(T )i = δi1...,irig(B ,B )...g(B ,B )
r j r! j1...,jrj i1j1 i2j2 ir−1jr−1 irjr
i1,...,iXr;j1,...,jr
for r even, and
(Tα)i = δi1...,irig(B ,B )...g(B ,B )(Aα)i
r j j1...,jrj i1j1 i2j2 ir−2jr−2 ir−1jr−1 j
i1,...,iXr;j1,...,jr
for r odd, where δi1,...,ir is the generalized Kronecker symbol and B = B(e ,e )
j1,...,jr ij i j
for some choice of orthonormal basis (e ). Then
i
1
S = (Tα )i(Aα)j for r even,
r r r 1 j i
−
i,j,α
X
1
S = (T )i(Aα)je for r odd,
r r r−1 j i α
i,j,α
X
denote the r–th symmetric functions of curvatures (for r odd the r–th mean cur-
vature is a vector field). In [10] the authors studied r–th minimality and stability
with respect to these generalized transformations. The stability conditions led to
non–existence results of stable minimal submanifolds on spheres [22, 10].
In this article, we define the notion of minimality associated with the general-
ized Newton transformation introduced by K. Andrzejewski, W. Koz lowski and
FRAME BUNDLE APPROACH TO GENERALIZED MINIMAL SUBMANIFOLDS 3
the author [3]. The idea of comes from then definition of symmetric functions
associated with a system A = (A ,...,A ) of matrices (endomorphisms),
1 q
χA(t) = σutu,
u
X
where u = (u ,...,u ) ∈ Nq is a multi–index, t = (t ,...,t ) and tu = tu1...,tuq.
1 q 1 q 1 q
Generalized Newton transformation T = T (A) depends on the multi–index u
u u
and a system of endomorphisms A and its recursive definition is the following
T0 = I, Tu = AαTα♭(u),
α
X
where α (u) is a multi–index obtained from u by subtracting 1 in the α–th coor-
♭
dinate. Moreover, it satisfies analogue of the condition (1.1), namely,
d d
σ (t) = tr A (t)·T .
dt u dt α α♭(u)
α (cid:18) (cid:19)
X
With the use of above characterization of T we study the variation of σ .
u u
In this setting, σ are symmetric functions of the family of shape operators
u
(Ae1,...,Aeq) corresponding to the choice of the orthonormal basis (e ) in the
α
normal bundle to L ⊂ M. Thus σ are functions on the principal bundle P of all
u
orthonormal frames over the submanifold. The integrals of σ over P with respect
u
to normalized measure are called total extrinsic curvatures and denoted by σˆ .
u
Notice, that in the case of distributions and foliations different types of cur-
vatures has been recently considered [21, 20], mainly, in order obtain integral
formulas. The ones used in this context have been introduced by K. Andrzejew-
ski, W. Koz lowski and the author [3] for distributions, not to study generalized
minimality, but to obtain different types of integral formulas related to extrinsic
geometry.
In this article, we derive the formula for the variation dσˆ . We show that at
dt u
t = 0
d
σˆ = g(R −S ,V)dvol ,
u u u L
dt
ZL
where V is the variation field and R ,S are integrals over the fibers of the
u u
following sections of the normal bundle to L over P,
R = tr((R(e ,·)e ) T )e , S = tr(A T )e ,
u α β ⊤ α♭(u) β u α u α
α,β α
X X
respectively. Thus the condition of u–minimality equals R = S (Theorem 5.6).
u u
In the case of M being a space form with sectional curvature c, R reduces to
u
c(n−|u|+1)H , where
u
Hu = σα♭(u)eα.
α
X
These results generalize the codimension one results by Reilly [17] and, in a sense,
the results of Cao and Li [10]. This is due to the fact that T in codimension one
u
reduces to the classical Newton transformation T , whereas transformations T in
r r
arbitrarycodimensionusedin[10]areobtainedfromT formulti–indices oflength
u
|u| = r (see [3]). In the end, we give some examples of u–minimal submanifolds.
4 K.NIEDZIAL OMSKI
Among others, we show that 0–minimality is equivalent to classical notion of
minimality.
Throughout the paper, we use the following index convention:
i,j,k = 1,2,...,dimL; α,β,γ = q +1,q +2,...,dimM.
2. Generalized Newton transformation and its basic properties
In [3] the authors introduced the notion of the generalized Newton transfor-
mation in order to study geometry of foliations of codimension higher than one.
This transformation generalizes the classical Newton transformation to the case
of finite family of operators. Let us recall this notion and state its properties.
Fix a positive integer q and let u ∈ Nq (N denotes here the set of all non–
negative integers). We define the length of u by |u| = u + u + ...,+u . Let
1 2 q
A = (A ,A ,...,A ) be a family of square n by n operators (matrices) on some
1 2 q
vector space V. Then we can consider the characteristic polynomial χA of the
form
χA(t) = det(I +tA),
where t = (t ,...,t ) and tA = t A +t A +...+t A . Clearly,
1 q 1 1 2 2 q q
χA(t) = σu(A)tu,
u n
|X|≤
for some constants σ = σ (A), where tu = tu1tu2...tuq for u = (u ,...,u ). We
u u 1 2 q 1 q
call σ the symmetric functions associated with the system A.
u
Definition 1. A system T = T (A), |u| ≤ n, of endomorphisms is called the
u u
generalized Newton transformation if the following condition holds: Let A(t) be
a one parameter family of operators such that A(0) = A and let σ (t) be the
u
corresponding symmetric functions. Then
d d
(2.1) σ (t) = tr A (t) ·T ,
dt u t=0 dt α t=0 α♭(u)
α (cid:18) (cid:19)
X
where α (u) denotes the multi–index obtained from u by subtracting 1 in the
♭
α–th coordinate.
One can show that (T ) exists and is unique [3]. Moreover, it can be
u u n
| |≤
characterized by the following recursive properties
T = I, 0 = (0,0,...,0),
0
(2.2) T = σ I − A T
u u α α♭(u)
α
X
= σ I − T A , |u| ≥ 1.
u α♭(u) α
α
X
The generalized Newton transformation has the following useful properties [3].
Proposition 2.1. Let A = (A ,...,A ) be a family of operators on a vector
1 q
space V. Let (T ) be the generalized Newton transformation associated with A
u
FRAME BUNDLE APPROACH TO GENERALIZED MINIMAL SUBMANIFOLDS 5
and let σ be the corresponding symmetric functions. Then the following relations
u
hold
|u|σ = tr(A T ),
u α α♭(u)
α
X
trT = (n−|u|)σ .
u u
One can obtain the direct formula for T and for σ (see [3], eq. (3) and (10)).
u u
In the Section 6 devoted to examples we will need the following algebraic
relation. Let us first recall necessary notation. For a multi–index u ∈ Nq and
p = |u| let
p p!
= .
u u !...u !
(cid:18) (cid:19) 1 q
Moreover, we write v ≤ u for two mulit–indices u,v,∈ Nq if u−v ∈ Nq.
Proposition 2.2. Let A be an endomorphisms of n–dimensional vector space
V. Let A = (ρ A + µ I,...,ρ A + µ I) be a system of endomorphisms, where
1 1 q q
ρ = (ρ ,...,ρ ) and µ = (µ ,...,µ ) are fixed elements of Rq. Then, for any
1 q 1 q
multi–index u ∈ Nq,
1 n−|u|+|v| |u|−|v|
(2.3) σ (A) = ρu vµvσ (A).
u − u v
(n−|u|)! v u−v | |−| |
v u(cid:18) (cid:19)(cid:18) (cid:19)
X≤
Proof. Assume first that A = (A +µ I,A ,...,A ) and A = (A ,A ,...,A )
1 1 2 q 0 1 2 q
for some endomorphisms A1,A2,...,Aq. Then the characteristic polynomial χA
equals
χA(t) = det(I +t1(A1 +µI)+t2A2 +...+tqAq)
= det((1+t µ)I +tA )
1 0
t
= (1+t µ)ndet I + A
1 1+t µ 0
(cid:18) 1 (cid:19)
= (1+t µ)n σ (A )(1+t µ) utu
1 u 0 1 −| |
u
X
= σ (A )(1+t µ)n utu
u 0 1 −| |
u
X
n u
−| | n−|u|
= σ (A ) µjtjtu.
u 0 j 1
u j=0 (cid:18) (cid:19)
X X
Thus
u1 n−|u|+j
σ (A) = µjσ (A ).
u j 1j(u) 0
♭
j=0 (cid:18) (cid:19)
X
Applying above formula consecutively to each ’coordinate’ we get
1 n−|u|+|v|
(2.4) σ (A +µ I,...,A +µ I) = µvσ (A ).
u 1 1 q q (n−|u|)! v u−v 0
v u(cid:18) (cid:19)
X≤
6 K.NIEDZIAL OMSKI
By the formula σ (A,A,A ,...,A ) = u1+u2 σ (A,A ,...,A ) (see
u 3 q u1 (u1+u2,u3,...,uq) 3 q
[21]) we have
(cid:0) (cid:1)
|u|
(2.5) σ (A,A,...,A) = σ (A).
u u
u | |
(cid:18) (cid:19)
Finally,(2.4),(2.5)andtheformulaσ (aA ,...,A ) = au1σ (A )imply(2.3). (cid:3)
u 1 q u 0
3. Frame bundle approach to submanifold geometry
Let (M,g) be a Riemannian manifold, L a codimension q immersed subman-
ifold of M, i.e., we have a surjective immersion ϕ : L → M. The Riemannian
metric on L is such that ϕ becomes an isometry and, therefore, we often identify
L with its image ϕ(L) ⊂ M. Consider the pull–back bundle ϕ 1TM over L with
−
the fiber
(ϕ 1TM) = T M, p ∈ L.
− p ϕ(p)
In other words, ϕ 1TM is a tangent bundle of M restricted to L. There is the
−
unique connection ∇ϕ in this bundle satisfying the following condition (see, for
example, [6])
∇ϕϕ Y = ∇ Y, X,Y ∈ Γ(TL),
X X
∗
where ∇ is the Levi–Civita connection on L.
The bundle ϕ 1TM splits into two bundles ϕ 1T L and ϕ 1T Lwith respect
− − ⊤ − ⊥
to the decomposition
T M = T ϕ(L)⊕(T ϕ(L)) , p ∈ L.
ϕ(p) ϕ(p) ϕ(p) ⊥
Denote by ∇ϕ, and ∇ϕ, the induced connections in these bundles, respectively.
⊤ ⊥
Notice that ϕ 1T L is isomorphic to TL via ϕ .
− ⊤
∗
ϕ 1T L ⊕ ϕ 1T L ϕ 1TM (cid:31)(cid:127) // TM
− ⊤ − ⊥ −
(cid:15) (cid:15)
(cid:15) (cid:15)
/ ϕ /
TL / L / M
Fix a local section N ∈ Γ(ϕ 1T L). Then N induces a shape operator AN :
− ⊥
TL → TL by the formula
ϕ AN(X) = (∇ϕN)⊤, X ∈ TL.
X
∗
Thus A ∈ Γ(End(TL)).
Let now P = O(ϕ 1T L) (or P = SO(ϕ 1T L) assuming L is transver-
− ⊥ − ⊥
sally orientable) be a bundle of (oriented) orthonormal frames of ϕ 1T L. Al-
− ⊥
ternatively, Γ(P) consists of all sections, which assign an orthonormal basis
(p,e) = (e ) , where e ∈ T L, to any point p ∈ L. For a fixed element
α α=1,...,q α ϕ⊥(p)
(p,e) of P we may consider a family of shape operators
A(p,e) = (Ae1,Ae2,...,Aeq).
Toeachsuchfamilywecanassociatethesymmetricfunctionsσ (p,e) = σ (A(p,e)),
u u
where u = (u ,u ,...,u ) ∈ Nq is a fixed multi–index. Thus σ ∈ C (P).
1 2 q u ∞
FRAME BUNDLE APPROACH TO GENERALIZED MINIMAL SUBMANIFOLDS 7
Each fiber P , p ∈ L, can be identified with the structure group G = O(q)
p
(or G = SO(q), respectively). Considering a (normalized) Haar measure on G we
define an integral of any smooth function f : P → R by
f dvol = f(e g)dg,
Px 0
ZPp ZG
where e is a fixed element of P . One can show that this definition is independent
0 p
of the choice of e . Therefore σ induces a smooth function σˆ on L,
0 u u
σˆ (p) = σ(p,e)dvol (p,e), p ∈ L.
u Pp
ZPp
We call σˆ the generalized extrinsic curvature of L or u–extrinsic curvature. For
u
some choices of u the generalized extrinsic curvatures vanish, but there are many
multi–indices uforwhichσˆ isnon–zeroandgives informationonthesubmanifold
u
(see the Section 7 – Final Remarks).
Moreover, let π 1TM be a vector bundle over P with the fiber
P−
(π 1TM) = T M.
P− (p,e) ϕ(p)
This bundle splits into direct (orthogonal) sum of bundles E and E with respect
′
to the decomposition T M = T L⊕(T L) . In other words,
ϕ(p) ϕ(p) ϕ(p) ⊥
E = (ϕ 1T L) , E = (ϕ 1T L) .
(p,e) − ⊤ p (′p,e) − ⊥ p
The connection ∇ϕ induces, with respect to the above decomposition, two con-
nections ∇E and ∇E′ in bundles E and E , respectively. In particular, we have
′
′
∇EWA αY = ∇EWAα Y −A∇EWeαY
′
(cid:0) (cid:1) = (cid:0)∇EW(AαY(cid:1))−Aα(∇EWY)−A∇EWeαY,
where we treat e as a section of E . Moreover, recall the Codazzi formula
α ′
(3.1) (Rϕ(X,Y)N) = (∇EA)NX −(∇EA)NY, X,Y ∈ TP, N ∈ E .
⊤ Y X ′
The key fact about differentiation of symmetric functions σ in the horizontal
u
direction follows by the definition (2.1) of generalized Newton transformation
and is given below. Denote by H the horizontal distribution in P induced by the
connection ∇ϕ, .
⊥
Proposition 3.1. The following formula holds
(3.2) Wσ = tr ∇E A ·T , W ∈ H.
u W α α♭(u)
α
X (cid:0)(cid:0) (cid:1) (cid:1)
Proof. Fix a basis (p,e) and let W ∈ H . Then W = Xh for some X ∈ T L.
(p,e) (p,e) p
Let γ be a curve on L such that γ(0) = p and γ˙(0) = X. Choose a local section
s of P parallel at (p,e). Then (see the Appendix) at (p,e) we may write
(∇EWA)αY = ∇ϕX,⊤(AeαY ◦s)−Aeα(∇ϕX,⊤(Y ◦s))−A∇ϕX,⊥eα(Y ◦s).
Consider a family B(t) = (B (t),...,B (t)) of endomorphisms of T L given by
1 q p
the formula
Bα(t) = (τt⊤)−1 ◦Aτt⊥(eα) ◦τt⊤,
8 K.NIEDZIAL OMSKI
where τ : T L → T L and τ : T L → T L denote the parallel transports
t⊤ p γ(t) t⊥ p⊥ γ⊥(t)
along γ in TL and T L, respectively. Since we compute dσ (γh(t)) , then
⊥ dt u t=0
we consider bases (e ) along γh(t). Therefore (e ) at γh(t) is a parallel trans-
α α
port of (e (p)). Hence τ (e (p)) = e (γh(t)) and B (t) = A (γh(t)). Moreover,
α t⊥ α α α α
dB (t) = (∇E A) , thus by (2.1) we obtain (3.2). (cid:3)
dt α t=0 W α
Let us define for a fixed multi–index u ∈ Nq three sections R ,H ,S ∈ Γ(E ).
u u u ′
Namely, we put
R (p,e) = − tr(R T )e ,
u αβ α♭(u) β
α,β
X
(3.3) H (p,e) = σ e ,
u α♭(u) α
α
X
S (p,e) = tr(A T )e ,
u α u α
α
X
where R : P → End(ϕ 1T L) is given by
αβ − ⊤
R (X) = (R(e ,X)e ) .
α,β α β ⊤
These sections induce sections R , H and S of ϕ 1T L, respectively, by
u u u − ⊥
R (p) = R (p) = R (p,e)dvol (p,e),
u u u Pp
ZPp
c
(3.4) H (p) = H (p) = H (p,e)dvol (p,e),
u u u Pp
ZPp
c
Su(p) = Su(p) = Su(p,e)dvolPp(p,e).
ZPp
The above integrals make secnse and are well defined (see the Appendix).
Remark 3.2. Notice that S is defined in the same way as S by Cao and Li
u r+1
[10], whereas we are not aware of the existence of the analogue of H (or H )
u u
in the literature, except for [3]. In [3] the authors computed divergence of the
section H +W , where
u u
Wu(p,e) = Tα♭β♭(u)(p,e)(∇eαeβ)⊤p .
α,β
X
and obtained the family of integral formulas, which generalize many formulas
relating intrinsic and extrinsic geometry of foliations known in the literature.
In the codimension one case,
(3.5) S = (u+1)σ N, and H = σ N,
u u u u 1
−
where N is a unit normal vector to L.
4. Properties of some differential operators
Consider the notation from the previous section. In this section we define two
differential operators, which will appear in the main considerations. These results
are an adaptation of Rosenberg’s results [19] concerning the classical Newton
transformation to the case of the generalized Newton transformation.
FRAME BUNDLE APPROACH TO GENERALIZED MINIMAL SUBMANIFOLDS 9
Firstly, for any section W ∈ Γ(E ) denote by W a function on P of the form
′ α
W (p,e) = g(W,e ) , (p,e) ∈ P.
α α p
Fix a muliti–index u ∈ Nq and consider a differential operator L : Γ(E ) →
u ′
C (P)
∞
L (W) = tr(((∇E)2W) ·T ), W ∈ Γ(E ).
u α α♭(u) ′
α
X
In other words,
L (W) = g((∇E)2 W),e )(T ) ,
u ei,ej α α♭(u) ji
α,i,j
X
where (e ) is an orthonormal basis on L and B denotes the coefficients of the
i ij
operator B : TL → TL with respect to this basis. Let us derive the formula for
the value of L on sections of the form V ◦ π , where V ∈ Γ(ϕ 1T L). Notice
u P − ⊤
that the definition of L is tensorial with respect to e . Thus, for a fixed point p,
u α
extending (e ) locally to a parallel basis at p, we may write L as
α u
L (V ◦π ) = g(∇E∇E (V ◦π ),e )
u P p ehi (Tα♭(u)ei)h P α
α,i
X
= g(∇ϕei,⊥∇ϕTα,⊥♭(u)eiV,eα)
α,i
X
= div(g(∇ϕTα,⊥♭(u)eiV,eα)ei),
α,i
X
where we also assume that (ei) is parallel at p. The quantity g(∇ϕTα,⊥♭(u)eiV,eα)ei
does not depend on the choice of orthonormal basis (e ) and, therefore, is well
i
defined section of E. In particular, it follows that (compare the Proposition 8.2
and the formula (8.2) in the Appendix)
(4.1) L (V ◦π )dvol = 0.
u P P
ZP
Since, for any function f ∈ C (L)
∞
g(∇ϕTα,⊥♭(u)eiV,eα)(eif) = g(∇ϕTα,⊥♭(u)(∇f)V,eα) = Tα♭(u)(∇f)g(V,eα),
α,i α α
X X X
we get
(4.2) fL (V ◦π )dvol = − (T (∇f))hV dvol .
u P P α♭(u) α P
ZP α ZP
X
We will compute the integral of the right hand side of (4.2). Let us begin
with preliminary results. These results are generalizations of the ones obtained
by Rosenberg [19].
Proposition 4.1. Let Y ∈ Γ(ϕ 1T L). The following relation holds
− ⊤
(4.3) g(∇E T (Y),e ◦π ) = g(∇E (Y ◦π ),e ◦π )+K (Y),
ehi u i P Tu(ei)h P i P u
i i
X X
10 K.NIEDZIAL OMSKI
where K (Y) is given by the following recursive relation
u
Ku(Y) = − Kα♭(u)(AαY)− g(Rϕ(Y,Tα♭(u)ei)eα,ei)− g(A∇ϕTα,⊥♭(u)eαY,ei)
α i,α i,α
X X X
and K (Y) = 0.
0
Proof. First, notice that the above relation is tensorial with respect to Y. Fix a
basis (p,e) ∈ P and consider a local section σ of P parallel at (p,e). Then the
formula (4.3) can be written at a point (p,e) as
g(∇ϕei,⊤(Tu(Y)◦σ),ei) = g(∇ϕTu,⊤(ei) σY,ei)+Ku(Y).
◦
i i
X X
We will use induction. Assume that the above formula holds for all multi–indices
v of length |v| < |u|. We may assume Y is parallel at p with respect to ∇ϕ, .
⊤
Thus, it suffices to show that
(4.4) g(∇ϕ, T (Y),e ) = K (Y).
ei⊤ u i u
i
X
By the recursive definition of T , Codazzi equation (3.1), formula (3.2) and in-
u
ductive assumption, we have
g(∇ϕ, T (Y),e ) = Y(σ ◦π )− g(∇ϕ, T A Y,e )
ei⊤ u i u P ei⊤ α♭(u) α i
i i,α
X X
= Y(σu ◦πP)− g((∇ϕTα,⊤♭(u)eiAα)Y,ei)− Kα♭(u)(AαY)
i,α α
X X
= Yhσ − g((∇ A) T e ,e )
u Y α α♭(u) i i
i,α
X
− g(Rϕ(Y,T e )e ,e )
α♭(u) i α i
i,α
X
− g(A∇Tϕα,⊥♭(u)eieαY,ei)− Kα♭(u)(AαY)
i,α α
X X
= K (Y),
u
(cid:3)
which proves (4.4).
In the Proposition below div denotes the divergence of sections of the bundle
E
E and Hess (ϕ) is a Hessian of ϕ ∈ C (P) in the horizontal direction (see the
∞
H
Appendix for more details).
Proposition 4.2. The following relation holds
div (T (∇f)) = tr(Hess (f ◦π )·T )+K (∇f), f ∈ C (L),
E u P u u ∞
H
for any multi–index u ∈ N(q).