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Let $M$ be a compact manifold and diff$(M)$ its diffeomorphism group. Let $G$ be a Lie subgroup of $GL(n,\mathbb{R})$. In general, the topology of a manifold may prevent it from having an integrable …

Let $M$ be a closed manifold, with dimension $2n+1$. Let $F(M)$ be the frame bundle, a principal $GL(2n+1,\mathbb{R})$-bundle over $M$. An almost CR structure $P$ on $M$ is a structure group reduction …

Let $M$ be an odd-dimensional manifold. An almost contact metric structure on $M$ is a 4-tuple $(\xi, \eta, \phi, g)$, where $\xi$ is a vector field, $\eta$ a one-form, $\phi$ an endomorphism of the t …

Let $M$ be a compact manifold and $\text{diff}(M)$ its diffeomorphism group. Various quotients of $\text{diff}(M)$ appear in the literature, oftentimes with geometric significance. A well-known exampl …

Let $M$ be a smooth manifold, of dimension $n$. I know that there are many types of geometric structures on $M$. A large number of them are captured by the notion of a $G$-structure, which is a reduct …

Consider the $n$-sphere $$ S^n = \{x\in\mathbb{R}^{n+1}: 1 - \sum_{k=1}^{n+1} x_k^2 = 0\}, $$ and let $g_1$ be the induced metric. Given $\lambda\in\mathbb{R}^{n+1}_{>0}$, we have the ellipsoid $$ E_\ …

Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. Rosenb …

It's well-known that on a Riemannian manifold $(M,g)$ with dimension larger than 2, the dimension of its conformal group $\text{conf}(M,g)$ is bounded above by ${n+2\choose 2}$. A Riemannian manifold …

Let $(M,g)$ be a Riemannian manifold of dimension $n$. Let $\text{conf}(M,g)$ denote the conformal group, i.e. the subgroup of diffeomorphisms of $M$ that acts by conformal transformations relative to …

Let $(M,g)$ be a compact manifold without boundary. Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known? An important example is the $n$-sphere …