On Carnot-Carathéodory metrics
On Carnot-Carathéodory metrics
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DOI：
10.4310/jdg/1214439462
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发表时间：
1985
影响因子：
2.5
通讯作者：
J. Mitchell
中科院分区：
文献类型：
--
作者：
J. Mitchell
考虑一个配备了\(k\) -平面的光滑分布的光滑黎曼\(n\) -流形\((M, g)\)。这样的一个分布\(\Delta\)给\(M\)中的每个点\(m\)分配切空间\(T_mM\)的一个\(k\)维子空间。\(M\)中的一条绝对连续曲线\(\alpha\)如果几乎处处与分布\(\Delta\)相切，则称它是水平的。可以如下在\(M\)上定义一个度量。
定义。\(M\)中两点\(p, q\)之间的卡诺 - 卡拉西奥多里距离\(d_C(p, q)=\inf_{\omega\in C}\{\text{length}(\omega)\}\)，其中\(C_{pq}\)是连接\(p\)到\(q\)的所有水平曲线的集合。如果分布\(\Delta\)满足霍尔曼德尔条件（假设\(M\)是连通的），则度量\(d_C\)是有限的。为了描述这个条件，设\(X_1, X_2,\cdots, X_k\)是\(M\)中\(m\)附近分布的向量场的一个局部基。如果这些向量场以及它们所有的换位子张成\(T_mM\)，那么就说这些向量场在\(m\)处满足霍尔曼德尔条件。用\(V_i(m)\)表示\(T_mM\)中由\(X_j\)的所有阶数\(< i\)的换位子张成的子空间（当然包括\(X_j\)本身）。很容易看出\(V_i(m)\)不依赖于局部基\(\{X_j\}\)的选择，所以如果对于某个\(i\)，\(\dim V_i(m)=\dim(M)\)，就说分布在\(m\)处满足霍尔曼德尔条件是有意义的。这种无穷小传递性意味着局部传递性：
定理（周）。如果一个光滑分布在\(M\)中的\(m\)处满足霍尔曼德尔条件，那么\(M\)中任何足够接近\(m\)的点\(p\)都可以通过一条水平曲线连接到\(m\)。因此，如果\(M\)是连通的，度量\(d_C\)是有限的。
我们将在下面证明关于与\(M\)上的一般分布\(\Delta\)相关联的度量空间\((M, d_C)\)的以下两个局部定理。（如果对于每个\(i\)，\(\dim(V_i(m))\)与点\(m\)无关，则称一个分布是一般的）
Consider a smooth Riemannian ^-manifold (M, g) equipped with a smooth distribution of /c-planes. Such a distribution Δ assigns to each point m e M a /^-dimensional subspace of the tangent space TmM. An absolutely continuous curve a in M is said to be horizontal if it is a.e. tangent to the distribution Δ. One may define a metric on M as follows. Definition. The Carnot-Caratheodory distance between two points p,q e M is dc{p, q) = infω e C {length(ω)}, where Cpq is the set of all horizontal curves which join p to q. The metric dc is finite provided that the distribution Δ satisfies Hόrmander's condition (assuming that M is connected). To describe this condition, let Xl9 X2, —,Xk be a local basis of vector fields for the distribution near m e M. If these vector fields, along with all their commutators, span TmM, then the vector fields are said to satisfy Hόrmander's condition at m. Denote by Vt{m) the subspace of TmM spanned by all commutators of the Λ̂ 's of order < i (including, of course, the X/s). It is easy to see that Vt{m) does not depend upon the choice of local basis {Xj}, so it makes sense to say that the distribution satisfies Hόrmander's condition at m if dimJ^(m) = dim(M) for some i. This infinitesimal transitivity implies local transitivity: Theorem {Chow). If a smooth distribution satisfies H'όrmander 's condition at m G M, then any point p e M which is sufficiently close to m may be joined to m by a horizontal curve. Thus, if M is connected, the metric dc is finite. We will prove below the following two local theorems concerning the metric space (M, dc) associated to a generic distribution Δ on M. (A distribution is said to be generic if, for each /, dim(P^(m)) is independent of the point