Category Archives: Differential Geometry

Let $\gamma: [0,1]\longrightarrow M$ be a path. Using connection $\nabla$, one can consider the notion of moving a vector in $L_{\gamma(0)}$ to $L_{\gamma(1)}$ without changing it. This is parallel transporting a vector from $L_{\gamma(0)}$ to $L_{\gamma(1)}$. The change is measured … Continue reading →

A connection on a line bundle can be defined in a pretty much similar fashion to a connection on a manifold that is discussed here since sections are like vector fields. Let $L\longrightarrow M$ be a line bundle. A connection … Continue reading →

A section of a line bundle is like a vector field. It is a map $s: M\longrightarrow L$ such that $s(m)\in L_m$ or $\pi\circ s(m)=m$. Section of a line bundle is one-to-one. Example. For the trivial bundle $L=M\times\mathbb{C}$,  every section … Continue reading →

Simply speaking, a line bundle is a complex vector bundle such that each fibre $F_x$ is a one-dimensional complex vector space i.e. one-dimensional vector space over the complex field $\mathbb{C}$. More specifically, Definition. A complex line bundle over a manifold … Continue reading →

Let $M$ be a differentiable manifold of dimension $n$. Consider an atlas $\mathcal{U}=\{U_\alpha\}_{\alpha\in\mathcal{A}}$ along with coordinates $x_\alpha^1,\cdots,x_\alpha^n$ in $U_\alpha$. For $x=(x_\alpha^1(x),\cdots,x_\alpha^n(x))\in U_\alpha$, a tangent vector is given by $$v=\sum_{j=1}^nv_\alpha^j\frac{\partial}{\partial x_\alpha^j}.$$ If $x\in U_\alpha\cap U_\beta$, then $v$ is also written as … Continue reading →

In $\mathbb{R}^n$, there is a globally defined orthonormal frame $$E_{1p}=(1,0,\cdots,0)_p,\ E_{2p}=(0,1,0,\cdots,0)_p,\cdots,E_{np}=(0,\cdots,0,1)_p.$$ For any tangent vector $X_p\in T_p(\mathbb{R}^n)$, $X_p=\sum_{i=1}^n\alpha^iE_{ip}$. Note that the coefficients $\alpha^i$ are the ones that distinguish tangent vectors in $T_p(\mathbb{R}^n)$. For a differentiable function $f$, the directional derivative … Continue reading →

A fibre bundle is an object $(E,M,F,\pi)$ consisting of The total space $E$; The base space $M$ with an open covering $\mathcal{U}=\{U_\alpha\}_{\alpha\in\mathcal{A}}$; The fibre $F$ and the projection map $E\stackrel{\pi}{ \longrightarrow}M$. The simplest case is $E=M\times F$. In this case, … Continue reading →