655
ISSN 1064–5624, Doklady Mathematics, 2008, Vol. 78, No. 2, pp. 655–659. © Pleiades Publishing, Ltd., 2008.
Original Russian Text © S.K. Vodop’yanov, M.B. Karmanova, 2008, published in Doklady Akademii Nauk, 2008, Vol. 422, No. 1, pp. 15–20.
In studying new metric objects, the need for area
formulas inevitably arises, which are applied in many
problems of analysis and applications. A noticeable
progress as compared with [1] was made in 1994, when
Kirchheim proved an area formula for Lipschitz map-
pings defined on Euclidean space and taking values in
an arbitrary metric space. In [2], this result was
extended to Lipschitz mappings defined on rectifiable
metric spaces and taking values in arbitrary metric
spaces (see also [3, 4], where the approaches of [5] to
studying metric structures and proving area formulas
were developed). Various area formulas were proved in
[6] for large classes of mappings, such as the Sobolev
classes [7] (and BV -mappings [8]) defined on
n
with
values in metric spaces.
In this paper, we prove an area formula for
C
1
-smooth contact mappings ϕ:
1
→
2
of Carnot–
Carathéodory spaces (or simply Carnot manifolds),
which naturally arise in physics, nonholonomic mechan-
ics, contact geometry, the theory of subelliptic equations,
tomography, neurobiology, robotics [9–11], and other
domains. The study of C
1
-smooth contact mappings is
of independent interest. The proof of the area formula
for such mappings contains a relation between Rieman-
nian and sub-Riemannian measures on the image sur-
face as an intermediate result; this makes it possible to
refine the area formulas obtained in [12, 13] for Lips-
chitz (with respect to the sub-Riemannian metric) map-
pings of Carnot groups, which are a special case of Car-
not manifolds.
Definition 1 (cf. [9]). Take a connected Riemannian
C
∞
-manifold of dimension N. The manifold is
called a Carnot manifold if the tangent bundle T has
a tangent subbundle H for which there exists a finite
set of positive integers dim H
x
= dimH
1
< dimH
2
< …
< dimH
i
< … < dimH
M
= N, where 1 < i < M, and each
point p ∈ has a neighborhood U ⊂ on which
C
1
-smooth vector fields X
1
, X
2
, …, X
N
are chosen so that,
at every point v ∈ U, the following conditions hold:
(i) X
1
(v), X
2
(v), … X
N
(v) form a basis in T
v
;
(ii) H
i
(v) = span{X
1
(v), X
2
(v), …, (v)} is a
subspace of dimension dim H
i
in T
v
;
(iii) [X
i
, X
j
](v) = (v)X
k
(v), where
the degree deg X
k
is defined as min{m| X
k
∈ H
m
};
(iv) the quotient mapping [·, ·]
0
: H
1
× H
j
/H
j – 1
H
j + 1
/H
j
(it is assumed that H
0
= {0}) induced by Lie
bracket is an epimorphism for all 1 ≤ j < M and v ∈ .
The number M is called the depth of the manifold .
Below, we define a quasi-metric, which we use in
what follows; we call it the sub-Riemannian quasi-
metric.
Definition 2 [14]. Let be a Carnot manifold with
topological dimension N and depth M, and let x =
exp X
i
(g). We define the quasi-distance d
2
(x, g) by
…
Definition 3. Suppose that g ∈ and (x
1
, x
2
, …,
x
N
) ∈ B
E
(0, r), where B
E
(0, r) is a Euclidean ball in
N
.
X
dimH
i
c
ijk
deg X
k
deg X
i
≤ deg X
j
+
∑
x
i
i 1 =
N
∑
⎝
⎜
⎛
⎠
⎟
⎞
d
2
xg , ( ) = max x
j
2
j 1 =
dimH
1
∑
⎝ ⎠
⎜ ⎟
⎛ ⎞
1
2
--
, x
j
2
j dimH
1
1 + =
dimH
2
∑
⎝ ⎠
⎜ ⎟
⎛ ⎞
1
2deg X
dimH
2
----------------------------
,
⎩
⎪
⎨
⎪
⎧
…, x
j
2
j dimH
M 1 –
1 + =
N
∑
⎝ ⎠
⎜ ⎟
⎛ ⎞
1
2deg X
N
-------------------
⎭
⎪
⎬
⎪
⎫
.
MATHEMATICS
An Area Formula for C
1
-Smooth Contact Mappings
of Carnot Manifolds
S. K. Vodop’yanov and M. B. Karmanova
Received March 21, 2008
Presented by Academician Yu.G. Reshetnyak March 13, 2008
DOI: 10.1134/S1064562408050037
Sobolev Institute of Mathematics, Siberian Branch, Russian
Academy of Sciences, pr. akademika Koptyuga 4,
Novosibirsk, 630090 Russia
e-mail: vodopis math.nsc.ru, maryka math.nsc.ru