Category Archives: Differential Geometry

Parallel Transport, Holonomy, and Curvature

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

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Sections of a Line Bundle II: Gauge Potential, Gauge Transformation, and Field Strength

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

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Sections of a Line Bundle I

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

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Line Bundles

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

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Vector Bundles

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

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Differentiable Manifolds and Tangent Spaces

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

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Fibre Bundles

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

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Structural Equations

Definition. The dual 1-forms $\theta_1,\theta_2,\theta_3$ of a frame $E_1,E_2,E_3$ on $\mathbb{E}^3$ are defined by $$\theta_i(v)=v\cdot E_i(p),\ v\in T_p\mathbb{E}^3.$$ Clearly $\theta_i$ is linear. Example. The dual 1-forms of the natural frame $U_1,U_2,U_3$ are $dx_1$, $dx_2$, $dx_3$ since $$dx_i(v)=v_i=v\cdot U_i(p)$$ for each … Continue reading

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Tensors I

Tensors may be considered as a generalization of vectors and covectors. They are extremely important quantities for studying differential geometry and physics. Let $M^n$ be an $n$-dimensional differentiable manifold. For each $x\in M^n$, let $E_x=T_xM^n$, i.e. the tangent space to … Continue reading

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Connection Forms

Let $E_1, E_2, E_3$ be an arbitrary frame field on $\mathbb{E}^3$. At each $v\in T_p\mathbb{E}^3$, $\nabla_v E_i\in T_p\mathbb{E}^3$, $i=1,2,3$. So, there exists uniquely 1-forms $\omega_{ij}:T_p\mathbb{E}^3\longrightarrow\mathbb{R}$, $i,j=1,2,3$ such that \begin{align*} \nabla_vE_1&=\omega_{11}(v)E_1(p)+\omega_{12}(v)E_2(p)+\omega_{13}(v)E_3(p),\\ \nabla_vE_2&=\omega_{21}(v)E_1(p)+\omega_{22}(v)E_2(p)+\omega_{23}(v)E_3(p),\\ \nabla_vE_3&=\omega_{31}(v)E_1(p)+\omega_{32}(v)E_2(p)+\omega_{33}(v)E_3(p) \end{align*} for each $v\in T_p\mathbb{E}^3$. These equations are … Continue reading

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