If you have two manifolds $M^m$ and $N^n$, how does one / can one decompose the diffeomorphisms $\text{Diff}(M\times N)$ in terms of $\text{Diff}(M)$ and $\text{Diff}(N)$? Is there …

I know the following facts. (Don't assume I know much more than the following facts.) The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bon …

I have often come across this implicit translation of the classical field of a particle of a given spin into a specific tensor field. But I could not locate any literature from whi …

Are Banach manifolds (or other types of infinite-dimensional manifolds) just curiosities, or have they been utilized to prove some interesting/important results? Where do they turn …

Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, …

This was asked as part of an earlier question. But since this part did not attract many answers, I am asking it separately. We consider the homology definition of an orientation f …

Poincaré Theorem on Kleinian groups (groups acting discontinously on Euclidean or hyperbolic spaces or on spheres) provides a method to obtain a presentation of a Kleinian group fr …

Real projective spaces ℝPn have ℤ/2 cohomology rings ℤ/2[x]/(xn+1) and total Stiefel-Whitney class (1+x)n+1 which is 1 when n is odd, so it follows that odd dimensional ones are bo …

So I have this manifold $M$, along with a metric $g_{\mu\nu}(x)$ and metric-compatible covariant derivative $\nabla_\mu$ (which is not necessarily the one corresponding to the Levi …

I've just finished my first course in differential geometry, so forgive me if this is maybe a silly or well-known question, but given any, say, diffeomorphism of $n$-manifolds $\ph …

Article Exotic $\mathbb{R}^4$ on Wikipedia says that there is at least one maximal smooth structure on $\mathbb{R}^4$, that is such an atlas on $\mathbb{R}^4$ that any other smooth …

Is there a finite dimensional closed manifold $M$ which is a $K(\pi,1)$, whose fundamental group is not word-hyperbolic, but which has a positive simplicial volume (ie "Gromov norm …

I am a beginner trying to learn about sheaves. I am trying to read Masaki Kashiwara's book "Sheaves on Manifolds", but I find it is not easy for me to understand. What other books …

Inspired by this thread, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topolo …