differential-geometry

The Bianchi identity is a differential equation satisfied by curvature data. It can be thought of as generalizing the equation for a real-valued 1-form to higher degree and nonabelian forms. Generally it applies to the curvature of ∞-Lie algebroid valued differential forms. Let be a smooth manifold. For a differential 1-form, its curvature 2-form is the de Rham differential . The Bianchi identity…

synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) differential equations, variational calculus Chern-Weil theory, ∞-Chern-Weil theory Cartan geometry (super, higher) Given a vector space and an elemen…

In measure theory one common thing we do is pulling back sigma algebras through functions which we want to force measurability for. Similar concept I have not heard for differentiable manifolds. That ...

At the moment, I am taking a course in differential geometry and am having some difficulty developing an intuition for the definition of an orientation on a manifold. I came across a couple of ...

I am considering a connected Riemannian manifold $M$ of dimension $n \geq 2$. Let $\gamma: [a, b) \to M$ (with $-\infty < a < b < \infty$) be a $C^\infty$ curve that is inextendible to the ...

Mathematics > Differential Geometry Title:A pictorial introduction to differential geometry, leading to Maxwell's equations as three pictures View PDFAbstract:In this article we present pictorially the foundation of differential geometry which is a crucial tool for multiple areas of physics, notably general and special relativity, but also mechanics, thermodynamics and solving differential equati…

I'm currently reading through Loring Tu's "An Introduction to Manifolds", which I've linked over here. I have a question about proposition 6.10, where Tu proves that every chart $(U, \phi)$ ...

synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) differential equations, variational calculus Chern-Weil theory, ∞-Chern-Weil theory Cartan geometry (super, higher) Let and be two smooth manifolds of…

See also differentiable function. analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeomet…

I would like to share a concise proof of the Poincaré Lemma using the flow of a radial vector field. Theorem (Poincaré Lemma) Assume that a smooth ##k##-form $$\omega = \sum_{i_1 < \dots < i_k} \omega_{i_1 \dots i_k} (x) dx^{i_1} \wedge \dots \wedge dx^{i_k}$$ is defined in a... Read more

Discussion of twisted differential K-theory and its relation to D-brane charge in type II string theory: and for KO: On differential Cohomotopy and Hypothesis H: On locality of extended functorial field theory: On the cobordism hypothesis for extended functorial field theory equipped with geometric structure (such as conformal field theory etc.): On the ordinary differential cohomology and differ…