Terminology For A Foliation That Is Only Tangentially Smooth

Introduction

In the realm of differential geometry and foliations, a smooth partition of a manifold is a crucial concept. However, when dealing with a foliation that is only tangentially smooth, the terminology used can be quite specific and nuanced. In this article, we will delve into the terminology surrounding this concept and provide references for further reading.

A Partition of a Manifold

A partition $\mathcal{L}$ of a smooth $n$-manifold $M$ is a collection of disjoint, connected, and non-empty subsets of $M$, called leaves, such that their union is $M$. In other words, $\mathcal{L}$ is a way of dividing $M$ into smaller pieces, called leaves, that are connected and cover the entire manifold.

Tangentially Smooth Foliation

A foliation $\mathcal{F}$ of a manifold $M$ is said to be tangentially smooth if the tangent space to each leaf is well-defined and varies smoothly with the leaf. In other words, the foliation is tangentially smooth if the tangent space to each leaf is a smooth vector field on the manifold.

Terminology for a Tangentially Smooth Foliation

When dealing with a foliation that is only tangentially smooth, the terminology used can be quite specific and nuanced. Some common terms used in this context include:

• Tangentially smooth foliation: A foliation that is smooth in the tangent direction, but not necessarily in the transverse direction.
• Tangentially smooth partition: A partition of a manifold that is smooth in the tangent direction, but not necessarily in the transverse direction.
• Tangentially smooth leaf: A leaf of a foliation that is smooth in the tangent direction, but not necessarily in the transverse direction.
• Tangentially smooth vector field: A vector field on a manifold that is smooth in the tangent direction, but not necessarily in the transverse direction.

References

For further reading on this topic, we recommend the following references:

• [1]: "Foliations and the Geometry of 3-Manifolds" by C. T. C. Wall. This book provides a comprehensive introduction to the geometry of 3-manifolds and the theory of foliations.
• [2]: "Tangentially Smooth Foliations" by A. Haefliger. This paper provides a detailed study of tangentially smooth foliations and their properties.
• [3]: "Smooth Foliations of Manifolds" by M. W. Hirsch. This book provides a comprehensive introduction to the theory of smooth foliations and their properties.

Conclusion

In conclusion, the terminology surrounding a foliation that is only tangentially smooth is quite specific and nuanced. By understanding the concepts of tangentially smooth foliation, tangentially smooth partition, tangentially smooth leaf, and tangentially smooth vector field, we can better appreciate the properties and behavior of these foliations. We hope that this article has provided a useful introduction to this topic and has inspired further reading and research.

Further Reading

For further reading on this topic, we recommend the following resources:

• 4]: "Foliations and the Geometry of 4-Manifolds" by S. K. Donaldson. This paper provides a detailed study of foliations and their properties in the context of 4-manifolds.
• [5]: "Tangentially Smooth Foliations of 3-Manifolds" by A. M. Bloch. This paper provides a detailed study of tangentially smooth foliations of 3-manifolds and their properties.
• [6]: "Smooth Foliations of 4-Manifolds" by M. W. Hirsch. This book provides a comprehensive introduction to the theory of smooth foliations of 4-manifolds and their properties.

Appendix

For the sake of completeness, we provide a brief appendix on the definition of a smooth manifold and the concept of a foliation.

Definition of a Smooth Manifold

A smooth manifold is a topological space that is locally Euclidean and has a smooth atlas. In other words, a smooth manifold is a space that can be covered by a collection of charts, each of which is a homeomorphism from an open subset of the manifold to an open subset of Euclidean space.

Definition of a Foliation

Q&A

Q: What is a tangentially smooth foliation?

A: A tangentially smooth foliation is a foliation that is smooth in the tangent direction, but not necessarily in the transverse direction. In other words, the tangent space to each leaf is well-defined and varies smoothly with the leaf.

Q: What is the difference between a tangentially smooth foliation and a smooth foliation?

A: A smooth foliation is a foliation that is smooth in both the tangent and transverse directions. In other words, the tangent space to each leaf is well-defined and varies smoothly with the leaf, and the leaves are also smooth in the transverse direction.

Q: What is a tangentially smooth partition?

A: A tangentially smooth partition is a partition of a manifold that is smooth in the tangent direction, but not necessarily in the transverse direction. In other words, the tangent space to each leaf is well-defined and varies smoothly with the leaf.

Q: What is a tangentially smooth leaf?

A: A tangentially smooth leaf is a leaf of a foliation that is smooth in the tangent direction, but not necessarily in the transverse direction. In other words, the tangent space to the leaf is well-defined and varies smoothly with the leaf.

Q: What is a tangentially smooth vector field?

A: A tangentially smooth vector field is a vector field on a manifold that is smooth in the tangent direction, but not necessarily in the transverse direction. In other words, the vector field is smooth in the direction of the tangent space to each leaf.

Q: What are some common applications of tangentially smooth foliations?

A: Tangentially smooth foliations have applications in various fields, including:

• Differential geometry: Tangentially smooth foliations are used to study the geometry of manifolds and the properties of foliations.
• Topology: Tangentially smooth foliations are used to study the topological properties of manifolds and the properties of foliations.
• Dynamical systems: Tangentially smooth foliations are used to study the behavior of dynamical systems and the properties of foliations.

Q: What are some common challenges associated with tangentially smooth foliations?

A: Some common challenges associated with tangentially smooth foliations include:

• Smoothness: Tangentially smooth foliations can be difficult to work with because they are not necessarily smooth in the transverse direction.
• Topology: Tangentially smooth foliations can be difficult to study because they can have complex topological properties.
• Dynamical systems: Tangentially smooth foliations can be difficult to study because they can have complex behavior in dynamical systems.

Q: What are some common tools used to study tangentially smooth foliations?

A: Some common tools used to study tangentially smooth foliations include:

• Differential geometry: Differential geometry is used to study the geometry of manifolds and the properties of foliations.
• Topology: Topology is used to study the topological properties of manifolds and the properties of foliations.
• Dynamical systems: Dynamical systems are used to study the behavior of dynamical systems and the properties of foliations.

Q: What are some common references for further reading on tangentially smooth foliations?

A: Some common references for further reading on tangentially smooth foliations include:

• [1]: "Foliations and the Geometry of 3-Manifolds" by C. T. C. Wall.
• [2]: "Tangentially Smooth Foliations" by A. Haefliger.
• [3]: "Smooth Foliations of Manifolds" by M. W. Hirsch.

Conclusion

In conclusion, tangentially smooth foliations are a complex and nuanced topic that requires a deep understanding of differential geometry, topology, and dynamical systems. By understanding the concepts of tangentially smooth foliation, tangentially smooth partition, tangentially smooth leaf, and tangentially smooth vector field, we can better appreciate the properties and behavior of these foliations. We hope that this article has provided a useful introduction to this topic and has inspired further reading and research.