स्नातकोत्तर प्रथम वर्षGeometry & Topologyनियोजित
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
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
41 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- 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.
परिभाषा
- 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.
प्रमेय
- 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.
आम भ्रांतियाँ
- Thinking every manifold comes with a preferred embedding in some ℝᴺ.
- Confusing vectors and covectors when changing coordinates.
वैकल्पिक खोजें
- 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
अधिगम पथ
आगे पढ़ने के लिए
- John M. Lee, Introduction to Smooth Manifolds