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To each second-order ordinary differential equation $\sigma $ on a smooth manifold $M$ a $G$-structure $P^\sigma $ on $J^1(\mathbb{R},M)$ is associated and the Chern connection $\nabla ^\sigma $ attached to $\sigma $ is proved to be…

Chern-Schwartz-MacPherson (CSM) classes generalize to singular and/or noncompact varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex manifold. The theory of CSM classes has been extended to…

If $X$ is a smooth manifold then the $\mathbb R$-algebra $C^\infty(X)$ of smooth functions $c:X\to\mathbb R$ is a $C^\infty$-$ring$. That is, for each smooth function $f:{\mathbb R}^n\to\mathbb R$ there is an $n$-fold operation…

We consider the pseudo-Riemannian Lichnerowicz conjecture in the homogeneous setting. In particular, we show that any compact connected pseudo-Riemannian manifold $M$ on which a semisimple group $G$ acts conformally, essentially and…

When $\mathcal{M}$ is a smooth, oriented, compact and simply connected manifold, Luc Menichi has shown that $HH^\ast(C^\ast(\mathcal{M}; \mathbb{F}))$, the Hochschild cohomology of the singular cochain complex of $\mathcal{M}$ is a…

Using the notion of isotopy modulo $k$, with $k \in \mathbb{N}^+$, we introduce a stratification on the set of all minimal $C_\infty$-algebra enhancements of a finite-type graded commutative algebra $H^*$. We determine obstruction classes…

Let $\pi:P\to B$ be a smooth $G$-bundle over a compact Riemannian manifold $B$ and $c$ a smooth loop in $B$ of constant seed $a(>0)$, where $G$ is compact semi-simple Lie group. In this paper, we prove that the holonomy map ${\rm…

Under some suitable assumptions Riemannian manifolds $(M, g, H)$ that admit a connection $\hat\nabla$ with torsion a 3-form $H$, which is both closed $d H=0$ and $\hat\nabla$-covariantly constant, are locally isometric to a product $N\times…

Let $M$ be a compact connected K\"ahler manifold and let ${\E}_{l-1}$ be the smallest term in the Harder-Narasimhan filtration of its tangent bundle. Let $G$ be an affine algebraic reductive group over $\C$. We prove the following result:…

For a differentiable manifold $M$, a pair $(M, \nabla)$ is called an affine manifold if $\nabla$ is a flat and torsion-free connection on the tangent bundle $TM\rightarrow M$. A Riemannian metric $g$ on $M$ is said to be a Hessian metric on…

Given an inclusion $A\hookrightarrow L$ of Lie algebroids sharing the same base manifold $M$, i.e. a Lie pair, we prove that the space $\Gamma(\Lambda^\bullet A^\vee)\otimes_{R} \frac{U(L)}{U(L)\cdot\Gamma(A)}$, where $R=C^\infty(M)$,…

If $\mathcal{M}=(M,\nabla)$ is an affine surface, let $\mathcal{Q}(\mathcal{M}):=\ker(\mathcal{H}+\frac1{m-1}\rho_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let…

Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V:=L(H). Using the algebraic structure of V, a torsionfree affine connection $\nabla$ (that is invariant under the group of…

Let $M$ be a finite volume analytic pseudo-Riemannian manifold that admits an isometric $G$-action with a dense orbit, where $G$ is a connected non-compact simple Lie group. For low-dimensional $M$, i.e. $\dim(M) < 2\dim(G)$, when the…

Given a complex Hilbert space H, we study the differential geometry of the manifold A of normal algebraic elements in Z=L(H), the algebra of bounded linear operators on H. We represent A as a disjoint union of subsets M of Z and, using the…

In this paper we consider the problem of identifying a connection $\nabla$ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian $\nabla^*\nabla$ over conformally transversally anisotropic…

For every compact almost complex manifold (M,J) equipped with a J-preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to…

We study the equation E_fc of flat connections in a fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an…

Using tools from the theory of Lie groupoids, we study the category of logarithmic flat connections on principal $G$-bundles, where $G$ is a complex reductive structure group. Flat connections on the affine line with a logarithmic…

The paper describes homomorphisms between Douglas algebras and some semisimple Banach algebras. The main tool is a result on the structure of the space $C(Z,\mathfrak M)$ of continuous mappings from a connected first-countable $T_1$ space…