The paper "Carnot-Caratheodory Spaces Seen from Within" by M. Gromov provides a comprehensive overview of Carnot-Caratheodory (C-C) geometry, focusing on the internal perspective of these spaces. The author introduces the concept of C-C metrics, which are defined using horizontal curves in a manifold \( V \) with a distinguished polarization \( H \subset T(V) \). These metrics are induced by an auxiliary Riemannian metric \( g \) and are used to measure distances between points in \( V \). Key topics covered in the paper include: 1. **Basic Definitions and Examples**: The paper begins with definitions of polarizations, horizontal curves, and C-C metrics. It discusses examples such as the Euclidean plane with piecewise linear curves and the Heisenberg group with a contact subbundle. 2. **Connectivity Theorems**: The Chow connectivity theorem is presented, showing that every two points in a manifold can be joined by a smooth \( H \)-horizontal curve. This theorem is extended to general Lie groups and the concept of self-similarity in C-C metrics. 3. **Hölder Equivalence and Mapping Problems**: The paper explores the Hölder equivalence between C-C metrics and Euclidean metrics, providing bounds on the Hölder exponent of maps between C-C spaces. It also addresses the Hölder mapping problem, which seeks to describe the space of \( C^\alpha \)-maps between C-C spaces. 4. **Hölder Surfaces and Homotopy Count**: The paper discusses the Hölder surfaces in contact 3-manifolds and the homotopy count of Hölder maps, providing bounds on the number of homotopy classes of Lipschitz maps between compact Riemannian manifolds. 5. **Asymptotic Geometry**: The asymptotic geometry of C-C spaces is explored, including the Pansu convergence theorem and the Mitchell cone theorem, which relate the asymptotic geometry of C-C spaces to that of nilpotent Lie groups. 6. **Internal and External Perspectives**: The paper highlights the distinction between internal and external perspectives in C-C geometry, emphasizing the importance of understanding the internal structure of C-C spaces to capture their essential characteristics. The paper aims to develop a robust internal C-C language and external analytic techniques to evaluate internal invariants of C-C spaces, particularly focusing on curves and hypersurfaces. It also addresses the challenges and open problems in the field, such as the precise formula for the exponent \( r \) in the homotopy count of Hölder maps.The paper "Carnot-Caratheodory Spaces Seen from Within" by M. Gromov provides a comprehensive overview of Carnot-Caratheodory (C-C) geometry, focusing on the internal perspective of these spaces. The author introduces the concept of C-C metrics, which are defined using horizontal curves in a manifold \( V \) with a distinguished polarization \( H \subset T(V) \). These metrics are induced by an auxiliary Riemannian metric \( g \) and are used to measure distances between points in \( V \). Key topics covered in the paper include: 1. **Basic Definitions and Examples**: The paper begins with definitions of polarizations, horizontal curves, and C-C metrics. It discusses examples such as the Euclidean plane with piecewise linear curves and the Heisenberg group with a contact subbundle. 2. **Connectivity Theorems**: The Chow connectivity theorem is presented, showing that every two points in a manifold can be joined by a smooth \( H \)-horizontal curve. This theorem is extended to general Lie groups and the concept of self-similarity in C-C metrics. 3. **Hölder Equivalence and Mapping Problems**: The paper explores the Hölder equivalence between C-C metrics and Euclidean metrics, providing bounds on the Hölder exponent of maps between C-C spaces. It also addresses the Hölder mapping problem, which seeks to describe the space of \( C^\alpha \)-maps between C-C spaces. 4. **Hölder Surfaces and Homotopy Count**: The paper discusses the Hölder surfaces in contact 3-manifolds and the homotopy count of Hölder maps, providing bounds on the number of homotopy classes of Lipschitz maps between compact Riemannian manifolds. 5. **Asymptotic Geometry**: The asymptotic geometry of C-C spaces is explored, including the Pansu convergence theorem and the Mitchell cone theorem, which relate the asymptotic geometry of C-C spaces to that of nilpotent Lie groups. 6. **Internal and External Perspectives**: The paper highlights the distinction between internal and external perspectives in C-C geometry, emphasizing the importance of understanding the internal structure of C-C spaces to capture their essential characteristics. The paper aims to develop a robust internal C-C language and external analytic techniques to evaluate internal invariants of C-C spaces, particularly focusing on curves and hypersurfaces. It also addresses the challenges and open problems in the field, such as the precise formula for the exponent \( r \) in the homotopy count of Hölder maps.