Isometric Embeddings of SubRiemannian Manifolds in R m Ovidiu Calin and Der-Chen Chang 1 Abstract: We discuss the subRiemannian geometry of codimension one with the subRiemannian met- ric induced by the Euclidean metric of R m . Key words: linear connection, subRiemannian geodesic, Euler-Lagrange equations MS Classification (2000): 53C17, 53C22, 35H20 1 Introduction This paper deals with the construction of an analog of the Gauss theory of hypersurfaces in the case of subRiemannain manifolds of codimension one. Readers may consult the articles [1] and [2] for more detailed discussion on subRiemannian geometry. Due to the non-commutativity, in the case of the subRiemannian geometry the second fundamental form coefficients are not symmetric, which leads to some modifications of the classical theory. An analog of the Levi- Civita connection is introduced, called adapted connection, see the works of Rumin, Fabel, Gorodski [5] and Webster [7]. The adapted Weingarten operator is introduced to model the way the distribution is shaped in the ambient space. The variational problem of subRiemannian geodesic is approached and the Euler-Lagrange equations are written in terms of the adapted connection and non-holonomic curvature. The last section deals with the example of the sphere S 3 , treated from the adapted connection point of view. See the articles [3] and [4] for other approaches of S 3 . Let D = span{X 1 ,...,X 2n } be a non-integrable distribution in R m , m =2n + 1, such that the subRiemannian metric g : Γ(D) × Γ(D) →F is induced from R 2n+1 by g(X i ,X j )= hX i ,X j i, ∀i, j, where h , i denotes the inner product on R m . Here Γ(D) is the set of all vector fields tangent to the distribution D. There is one missing direction and it is normal to the distribution D at each point of the space. This has the same direction with the normal unit vector field N defined by hN,X i i =0, hN,N i =1. Let N = ∑ ω i e i be the coordinate representation of the unit normal vector. Then the one-form ω = ∑ ω i dx i annihilates the distribution D and ω(N )= |N | 2 = 1. Let ∇ denote the Levi-Civita connection on R 2n+1 , which is given by ∇ U V = m X k=1 U (V k )e k , 1 The first author partially supported by a research grant at Eastern Michigan University. The second author partially supported by a competitive research grant at Georgetown University and a grant from United States Army Research Office. 1