Differential Geometry

Given a principal \(G\) bundle \(P\rightarrow M\) we associate what is called a Weil homomorphism \(I(G)\rightarrow H^*(M,\mathbb{R})\). Given \(f\in I^k(G)\) i.e., \(f:\underbrace{\mathfrak{g}\times\cdots\times\mathfrak{g}}_{k\text{ times}}\rightarrow \mathbb{R}\) we associate an element in \(H^{2k}(M,\mathbb{R})\) as follows. This is only an outline. It is useful if you can fill the gaps by your self. Fix a connection \(\Gamma\) on \(P(M,G)\) and let \(\Omega\) denote the curvature form associated to \(\Gamma\). The element \(f\in I^k(G)\) gives a \(2k\)-form \(f(\Omega):P\rightarrow \Lambda^{2k}T^*P\) on \(P\) as follows. \(f(\Omega)(v_1,\cdots,v_{2k})=\frac{1}{(2k)!}\sum_{\sigma\in S_{2k}} f(\Omega(v_{\sigma(1)}.v_{\sigma(2)}),\cdots\Omega(v_{\sigma(2k-1)},v_{\sigma(2k)}))\) ...

Let \(G\) be a Lie group and \(\mathfrak{g}\) be its Lie algebra. Let \(P\rightarrow M\) be a principal \(G\) bundle. A connection form on \(P\) is a \(\mathfrak{g}\) valued \(1\)-form on \(P\) satisfying some properties. Suppose \(G=Gl(n,\mathbb{R})\) then \(\mathfrak{g}=M(n,\mathbb{R})\). A connection is given by \(\omega:P\rightarrow \Lambda^1_{\mathfrak{g}}T^*P\). Given \(p\in P\) we have \(\omega(p):T_pP \rightarrow \mathfrak{g}\). Given \(v\in T_pP\), \(\omega(p)(v)\) is a matrix \((a_{ij})\in M(n,\mathbb{R})\) i.e., given \(v\in T_pP\) we have \(n^2\) real numbers \(a_{ij}\in \mathbb{R}\) associated to it. Varying \(v\) over \(T_pP\) gives \(n^2\) maps \(a_{ij}:T_pP\rightarrow \mathbb{R}\). So, given \(p\in P\), we have \(n^2\) maps \(\omega_{ij}(p):T_pP\rightarrow \mathbb{R}\) where \(\omega_{ij}(p)(v)\) is the \(ij\) th component of \(\omega(p)(v)\). Fix \(i,j\) then, \(\omega_{ij}:P\rightarrow \Lambda^1 T^*P\) given by \(p\mapsto \omega_{ij}(p)\) is a real valued \(1\)-form on \(P\). Thus, we denote \(\omega\) by \((\omega_{ij})\) where \(\omega_{ij}\) are real valued \(1\)-forms on \(P\). This is what it means to see connection as a matrix of \(1\)-forms. The same can be done for Curvature form also. Curvature form \(\Omega:P\rightarrow \Lambda^2_{\mathfrak{g}}TP\) associates for each \(p\in P\) a map \(\Omega(p):T_pP\times T_pP\rightarrow \mathfrak{g}\). Same explanation as above gives \(n^2\) real valued \(2\)-forms \(\Omega_{ij}:P\rightarrow \Lambda^2 TP\). We denote Curvature form \(\Omega\) by \((\Omega_{ij})\). This is what it means to see curvature as a matrix of \(2\)-forms.

Here, I will add links for web pages where I have written about concepts from Kobayashi and Nomizu's book Foundations of Differential geometry (Volume \(1\) and Volume \(2\)). Derivative of Left invariant differential form Maurer-Cartan form on a Lie group Transition maps for principal bundle are smooth Trivializations and sections in Principal bundle Construction of Weil homomorphism Construction of associated bundle Equivariant maps are Isomorphisms Invariant polynomials

In this, we see that for a left-invarinat differential form \(\omega\) on $G$, \(d\omega(X,Y)=-\frac{1}{2}\omega([X,Y])\) for vector fields \(X, Y\) in \(G\). Let \(\omega:G\rightarrow \Lambda^1 T^*G\) be a Left-invariant differential form on \(G\) i.e., \((L_g)^*\omega=\omega\) for all \(g\in G\) i.e., \(\omega(g)(v)=\omega(e)((L_{g^{-1}})_{*,g}(v))\) for all \(g\in G\) and \(v\in T_gG\). Given \(A\in \mathfrak{g}\) we have vector field \(A^*:G\rightarrow TG\) defined as \(A^*(g)=(L_g)_{*,e}(A)\). Then, \(\omega(A^*):G\rightarrow \mathbb{R}\) is constant function, \(\omega(A^*)(g)=\omega(e)(A)\) for all \(g\in G\). As \(\omega(A^*):G\rightarrow \mathbb{R}\) is constant map, \(X( \omega(A^*))=0\) for any vector field \(X:G\rightarrow TG\) on \(G\). In particular, \(B^*(X(\omega(A^*)))=0\) for \(B\in \mathfrak{g}\). Interchanging \(A\) and \(B\) we have \(A^*(\omega(B^*))=0\) As \((d\omega)(A^*,B^*)=\frac{1}{2}\left[ A^*(\omega(B^*))-B^*(\omega(A^*))-\omega([A^*,B^*])\right]\) ...

Let \(G\) be a Lie group and \(\mathfrak{g}\) be its Lie algebra. We want to associate a \(\mathfrak{g}\) valued \(1\) form on \(G\). We define \(\theta:G\rightarrow \Lambda^1_{\mathfrak{g}}T^*G\) as follows. For \(g\in G\), we need \(\theta(g):T_gG\rightarrow \mathfrak{g}=T_eG\). For manifolds \(M,N\), one natural way to get a map between tangent spaces \(T_mM\) and \(T_nN\) is to think of a smooth map \(f:M\rightarrow N\) such that \(f(m)=n\) and take its differential at \(m\). We get \(f_{*,m}:T_mM\rightarrow T_nN\). To get \(\theta(g):T_gG\rightarrow \mathfrak{g}=T_eG\), we look for a map \(G\rightarrow G\) that takes \(g\) to \(e\). One such map is multiplication by \(g^{-1}\). Consider \(\delta_{g^{-1}}:G\rightarrow G\) given by \(h\mapsto g^{-1}h\). This map takes \(g\) to \(e\) and \(\delta_{*,g^{-1}}:T_gG\rightarrow T_eG=\mathfrak{g}\). This gives a \(\mathfrak{g}\) valued \(1\)-form on \(G\) which we call to be the Maurer-Cartan form on \(G\) denoted by \(\theta\) defined as \(\theta(g)=(\delta_{g^{-1}})_{*,g}:T_gG\rightarrow \mathfrak{g}\). Kobayashi and Nomizu defines Maurer-Cartan form on \(G\) to be "the left-invariant \(\mathfrak{g}\) valued \(1\)-form on \(G\) uniquely determined by the condition that \(\theta(A)=A\) for all \(A\in \mathfrak{g}\). More precisely, this means \(\theta (e) :T_eG\rightarrow T_eG\) is such that \(\theta(e)(A)=A\) for all \(A\in \mathfrak{g}\). The condition that \(\theta\) is left-invariant means that \(\theta(e):T_eG\rightarrow \mathfrak{g}\) determines what \(\theta(g):T_gG\rightarrow \mathfrak{g}\). So, the condition \(\theta(e)(A)=A\) for all \(A\in \mathfrak{g}\) along with the condition left-invariant gives unique \(1\)-form \(\theta\) which is called as the Maurer-Cartan form. Once we unravel how \(\theta(e):T_eG\rightarrow \mathfrak{g}\) determines \(\theta(g):T_gG\rightarrow \mathfrak{g}\) and that \(\theta(e)(A)=A\) for all \(A\in \mathfrak{g}\) we see that \(\theta(g)(v)=(\delta_{g^{-1}})_{*,g}(v)\) for all \(v\in T_gG\) which is precisely what I have written in the first half of this post. Differentiating this Maurer-Cartan form \(\theta:G\rightarrow \Lambda^1_{\mathfrak{g}}T^*G\) we see that \(d\theta(X,Y)=-\frac{1}{2}\theta([X,Y])\) ...

I am Praphulla Koushik, faculty member in department of Mathematics, NIT Calicut. I am interested broadly in Differential geometry and Category theory. I want to share my thoughts as I read the following books. Hartshorne's Algebraic geometry. ... ... I wish to add some articles that seems to be interesting for me on different topics. Most of the times, I start writing a post and leave it in the middle.. I usually write in usual latex before copying it here. It is time consuming to put latex in between dollars. I have to take screen shot of figures, crop into image and then add here which is really time consuming. So, If I feel I am wasting time, I will leave it like that. If you find any of my posts interesting and incomplete, do leave a message. I will continue it.