研究生一年级Geometry & Topology计划中
Smooth manifolds
Smooth manifolds, tangent and cotangent bundles, vector fields and flows, differential forms, de Rham cohomology, Stokes's theorem and an introduction to Riemannian metrics.
math.geom.smooth-manifolds
本主题的课程尚未编写。下面的提纲是课程地图的一部分，尚未经过审校。
前置知识
41 项必需的前置知识（一直追溯到底）
你将学会
- Show that spheres, projective spaces and matrix groups are smooth manifolds, using the regular value theorem.
- Compute with tangent vectors, vector fields, differential forms and pullbacks.
- Prove Stokes's theorem for manifolds with boundary and compute the de Rham cohomology of simple manifolds.
定义
- A smooth n-manifold is a second-countable Hausdorff space with a maximal atlas of charts to ℝⁿ whose transition maps are smooth.
- The tangent space TₚM can be defined as the space of derivations at p.
- A differential k-form is a smooth section of the k-th exterior power of the cotangent bundle.
定理
- Regular value theorem: the preimage of a regular value of a smooth map is an embedded submanifold.
- Stokes: ∫M dω = ∫{∂M} ω for a compactly supported (n − 1)-form ω on an oriented n-manifold M with boundary.
- Whitney: every smooth n-manifold embeds in ℝ²ⁿ.
- de Rham: de Rham cohomology is isomorphic to singular cohomology with real coefficients.
常见误解
- Thinking every manifold comes with a preferred embedding in some ℝᴺ.
- Confusing vectors and covectors when changing coordinates.
可选发现
- Functors The tangent construction is a functor from smooth manifolds to vector bundles, and the chain rule is precisely the statement that it preserves composition.
- ← Classical mechanics
- ← Differential geometry of curves and surfaces
学习路径
延伸阅读
- John M. Lee, Introduction to Smooth Manifolds