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Bott & Tu in Differential forms in Algebraic Topology write in Remark 5.17, pg.52 The two Poincare duals of a compact orientated submanifold correspond to two homology theories - closed and compac …

In the book Foundations of Differential Geometry by Kobayashi & Nomizu, the fundamental vector field is defined in section 4, pg.42: If $G$ acts on $M$ on the right, we assign to each element $X\i …

The Chern-Gauss-Bonnet theorem can be deduced from the Atiyah-Singer theorem by applying it to the differential operator $d+d^*$ mapping from sum of even powers of the exterior sheaf to the sum of odd …

On the Woldram Mathworld site, tgs Jacobi tensor is defined to be: $J^m_{ijk} :=1/2( R^m_{ijk} + R^m_{kji})$ where $R^m_{ijk}$ is the Riemann curvature tensor. I've not been able to find anything else …

Although higher derivatives are best thought through with jet bundles where we actually differentiate a bundle and not just a manifold, there is a useful description of second derivatives using second …

In a paper by Rafael Dahman on the embedding of the long line into weakly complete vector spaces, those of the form $R^I$ for arbitrary $I$, and equipped with the Michal-Bastiani calculus, he notes a …

Most of the references I've seen deal with Riemannian geometry, rather than semi-Riemannian geometry. Chens monograph, Pseudo-Riemannian Geometry, $\Delta$-Invariants and Applications is one of the fe …

According to an article in The Geometry and Topology of Submanifolds and Currents, edited by Weiping Li and Shinshu Walter Wei The lack of suitable representatives in smooth homology by submanifolds …

A solenoidal tangent field, mathematically speaking, is one whose divergence vanishes. They are also called incompressible. I understand why they are called incompressible — a fluid flow is called inc …

I'm currently reading Michor, Manifold of Mappings for Continuum Mechanics. In this paper he makes use of 'Whitney Jets' but takes it to be an already understood concept. I'm familiar with jets but ha …

Personally, I think the best way to illustrate curvature is to start from the simplest case. This is how Hilbert illustrated it in his book, Geometry and the Imagination. The simplest curve is the t …

The only exposition of de Rham homology I've found is an appendix to Uranga and Ibanezs book on String Phenomenology. It was brief and gave only basic outline of how to construct this homology. Now de …

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\End{End}$ Todorov and Dubois-Violette have recently shown how to understand the structural gauge group of the standard model via octonions. Q. Is there …

Have you considered the curves of constant width? These are curves where the minimum distance between two parallel and tangential lines are the same no matter what the orientation of the line is. Thes …

There's a discussion of the history of the Godbillon-Vey invariant in Hurder's Dynamic's & the Godbillon-Vey Class. He recalls: Reinhart and Wood ... gave a formula expressing the the Godbillon-Vey c …