Geometry and some category theory

In this note we collect some references that discuss the notion of connection on vector bundle Differential geometry by Loring Tu Geometry of Differential forms by Shigeyuki Morita Global Calculus by S Ramanan From Calculus to Cohomology by Madsen Natural Operations in differential geometry by Kolar, Michor, Slovak Foundations of Differential geometry by Kobayashi and Nomizu Differential geometry by Taubes Geometry of Physics by Theodore Frankel Modern differential geometry for Physicists by Chris Isham Differential Geometry by Loring Tu ...

Let \(\pi:P\rightarrow M\) be a principal \(G\) bundle. Let \(F\) be a smooth manifold with an action of \(G\) from left (note that action of \(G\) on \(P\) is from right). Given this we want to associate a fiber bundle over \(M\). This action is same thing as giving a smooth map \(G\times F\rightarrow F\). We look for a fiber bundle with fibre \(G\times F\) and see if we can construct another fibre bundle with fibre \(F\) from the map \(G\times F\rightarrow F\). ...

We will understand the notion of a connection on a vector bundle in the following steps: Give the definition of a connection Explain the objects appearing in the definition (sections and their algebraic structure) Study the trivial bundle case, which motivates the axioms Examine the tangent bundle case and test familiar operations Explain why the usual differential of a section does not give what we want Let \(E\rightarrow M\) be a vector bundle. ...

Let \(M\) be a skew-symmetric matrix (with real entries). Let \(\lambda\) be an eigenvalue of \(M\). This means, there exists vector \(v\) such that \(Mv=\lambda v\). To relate with ``skew-symmetricness'' of \(M\), we apply transpose on both sides of previous equation, to get \(v^TM^T=\lambda v^T\). As \(M\) is skew-symmetric, we see that \(v^TM^T=\lambda v^T\) is equivalent to \(-v^TM=\lambda v^T\). Now, multiply by \(v\) on both sides of the above equation to get \(-v^TMv=\lambda v^Tv\). ...

On a Saturday morning, I was thinking about sylow theorems. The question I asked myself is, do I know how to apply sylow theorems? Only application I was aware about, of sylow theorem, is to assure if a group of finite order is simple or not. As a first step, I thought to check for groups of order less than 100. ...

Let us consider a problem where you are asked to find infimum of the set \[\{\int_0^{1}\sqrt{1+f'(x)^2}dx\}_{f\in S}\] where \(S\) is the set of all \(f\in C^1(\mathbb{R})\) with the property that \(f(0)=10\) and \(f(1)=0\). When we see integral and differential together, that should remind us the famous fundamental theorem of calculus, which says that ...

Let us check for limit/limsup/liminf of the sequence \(\frac{n}{10^{\lceil \log_{10}n \rceil}}\), where the notation \(\lceil x \rceil\) means the smallest integer greater than or equal to \(x\). For example, \(\lceil 0.1 \rceil=1, \lceil 0.9 \rceil=1, \lceil -1.2 \rceil=-1, \lceil -2.5 \rceil=-2\) To compute limit (to have a hope of computing), we need to know it converge (which we can check by checking it is Cauchy sequence). ...

Let \(M\) be a smooth manifold and \(E\rightarrow M\) a vector bundle over \(M\). A connection on the vector bundle \(E\rightarrow M\) is usually defined as a map \[\nabla : \Gamma(M,TM)\times \Gamma(M,E)\rightarrow \Gamma(M,E)\] satisfying the following conditions: \(\nabla\) behaves very well with the \(\mathbb{R}\)-vector space structure on \(\Gamma(M,TM)\) and \(\Gamma(M,E)\); in the sense that, \(\nabla\) is an \(\mathbb{R}\)-bilinear map, \(\nabla\) behaves reasonably well with the \(C^\infty(M)\)-module structure on \(\Gamma(M,TM)\) and \(\Gamma(M,E)\); in the sense that, \[\nabla(fX,s)=f\nabla(X,s)\] for \(X\in \Gamma(M,TM)\) and \(s\in\Gamma(M,E)\) ...

Let us look at the first class of multilinear maps; the bilinear maps. We want to study bilinear maps. The notion of "study" will have different meanings as we move forward (or backward) in the course. Let \(V,W,T\) be vector spaces and \(\varphi:V\times W\rightarrow T\) be a bilinear map. The feeling that "we are good at linear algebra" suggests us to ask the question : ...

In group theory, we mainly study maps that preserve the group structures; which goes by the name of group homomorphisms. In topology, we mainly study maps that preserve the topologies; which goes by the name of continuous functions. In theory of vector spaces, we mainly study maps that preserve the vector space structures; which goes by the name of linear maps. Apart from that, there are many interesting maps that comes up when dealing with vector spaces which are not really linear maps. The very first example that comes to mind is the determinant map ...