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Let $U$ be an open set in $\mathbb{R}^n$ (or more generally, a manifold) and let $V$ be a separated bornological vector space. Do we have $$C^\infty(U, V) \cong C^\infty(U) \,\hat{\otimes}\, V,$$ as b …

In Kriegl/Michor's book "The convenient setting of global analysis", they define the space of differential $k$-forms on a possibly infinite-dimensional manifold $M$ as the space of smooth sections of …

For appropriate choices of $i$, $j$ (e.g. $i+j \neq n$), $\Omega^i(M)$ and $\Omega^j(M)$ have different ranks as $C^\infty(M)$ module, so at least they cannot be isomorphic as modules. Maybe they coul …

Yes, there is indeed a mistake. Chavels Lemma 2 on page 153 tells you that $$L(H_k * F) = (LH_k)*F - F,$$ so if you define $F = \sum_{l=1}^\infty (LH_k)^{*l}$ and $p= H_k + H_k * F$, then $$ L p = LH_ …

In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form $$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$ where $X$ is a semimartingale on a manifold $M …

In Chern-Weil Theory, characteristic classes appear as certain closed differential forms associated to vector bundles with connections (constructed via the curvature form of the connection). When yo …

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet s …

I wonder that this was not said before. Take a triangulation of your manifold. Choose disjoint open balls around each 0-cell in the triangulation and set the union to be $U_0$ (a ball means here some …

In the Wikipedia article on Hilbert manifolds, it is claimed that every Hilbert manifold can be smoothly embedded onto an open subset of the model Hilbert space. However, no explicit reference is give …

Given a Dirac operator $D$ acting on some Clifford bundle $\mathcal{E}$ over a compact, even-dimensional, oriented manifold $M$, the Atiyah-Singer index theorem states that its index is given by pairi …

The method of stationary phase is very well-known and employed in many areas of physics and mathematics, and, of course, included in various versions as theorem in textbooks, especially on pseudors an …

Let $X$ be a vector field on a compact manifold $M$ that has the form $$ X = \lambda_1 x^1 \partial_1 + \dots + \lambda_n x^n \partial_n + \dots$$ with respect to some chart $x$ around a point $p$. Al …