🌟 Smooth Manifolds Seminar — Tangent Spaces of Manifolds
Date:
The undergraduates at Hua Loo-Keng Talent Program in Mathematics organized a seminar on smooth manifolds after learning topology and basic differential geometry during 2025 winter vacation. I gave 2 lectures in this seminar and the following is the second lecture on tangent spaces of smooth manifolds.
I first introduced the opinion of smooth category, with objects as smooth manifolds and morphism as smooth maps, before introducing the differential functor from smooth category to $\mathbb{R}$-linear space category. In detial, I first introduced the tangent space of a hypersurface in $\mathbb{R}^n$ and revealed the relationship between direction vectors $\mathbf{v}$, direction derivatives $\dfrac{\partial f}{\partial \textbf v}$ and derivation $D_{\textbf{v}}$, which would be generalized and applied to the concepts of smooth manifolds. After introducing the tangent space of smooth manifolds, I also discussed submersion and immersion maps and the rank theorem of smooth maps.
⏬ Click to download the hand-written SEMINAR NOTES of the research paper in PDF.