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We prove the following statement: Let g be a light-line-complete pseudo-Riemannian Einstein metric of indefinite signature on a connected (n>2)-dimensional manifold M. Assume that a conformally equivalent metric is also Einstein. Then, the…

In this paper, we study the deformed Hermitian-Yang-Mills equation on compact K\"ahler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with…

Let $M$ be a compact complex manifold admitting a K\"ahler structure. A conformally K\"ahler, Einstein-Maxwell metric (cKEM metric for short) is a Hermitian metric $\tilde{g}$ on $M$ with constant scalar curvature such that there is a…

In this paper, we investigate the rigidity of Q-curvature. Specifically, we consider a closed, oriented $n$-dimensional ($n\geq6$) Riemannian manifold $(M,g)$ and prove the following results under the condition $\int_{M} \nabla R\cdot\nabla…

We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric…

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has at most $C$-quadratic decay at infinity for some $C > \frac{2}{3}$, then it decomposes as a (possibly infinite)…

We are interested in the geometry of the group $\mathcal{D}_q(M)$ of diffeomorphisms preserving a contact form $\theta$ on a manifold $M$. We define a Riemannian metric on $\mathcal{D}_q(M)$, compute the corresponding geodesic equation, and…

We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the "link" of the…

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 complete flat pseudo-Riemannian homogeneous manifold and $\Gamma\subset\Iso(\RR^n_s)$ its fundamental group. We show that $M$ is a trivial fiber bundle $G/\Gamma\to M\to\RR^{n-k}$, where $G$ is the Zariski closure of $\Gamma$ in…

We generalise Hinich's Theorem of descent of Deligne groupoids to the case where the dgLas involved have no negative cohomology. We apply this result to study the infinitesimal deformations of a morphism $\alpha: {\mathcal F} \to {\mathcal…

We obtain a coordinate independent algorithm to determine the class of conformal Killing vectors of a locally conformally flat $n$-metric $\gamma$ of signature $(r,s)$ modulo conformal transformations of $\gamma$. This is done in terms of…

Let $X$ be a compact connected Riemann surface of genus $g$, with $g \geq 2$. For each $d <\eta(X)$, where $\eta(X)$ is the gonality of $X$, the symmetric product $\text{Sym}^d(X)$ embeds into $\text{Pic}^d(X)$ by sending an effective…

By a famous result of K. Saito, the parameter space of the miniversal deformation of the $A_{r-1}$-singularity carries a Frobenius manifold structure. The Landau-Ginzburg mirror symmetry says that, in the flat coordinates, the potential of…

On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\mb{S}^{\tn=\tn(n)},\tg)$, we consider the total exterior scalar curvature $\Theta_{f_g}(M)$, and squared $L^2$ norm of the mean curvature…

We relate $L^p$ convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal…

$(N(k),\xi)$-semi-Riemannian manifolds are defined. Examples and properties of $(N(k),\xi)$-semi-Riemannian manifolds are given. Some relations involving ${\cal T}_{a}$-curvature tensor in $(N(k),\xi)$-semi-Riemannian manifolds are proved.…

To any non-negatively graded dg Lie algebra $g$ over a field $k$ of characteristic zero we assign a functor $\Sigma_g: art/k \to Kan$ from the category of commutative local artinian $k$-algebras with the residue field $k$ to the category of…

We discuss the geometry of Laplacian eigenfunctions $-\Delta \phi = \lambda \phi$ on compact manifolds $(M,g)$ and combinatorial graphs $G=(V,E)$. The 'dual' geometry of Laplacian eigenfunctions is well understood on $\mathbb{T}^d$…

A definition of space-time metric deformations on an $n$-dimensional manifold is given. We show that such deformations can be regarded as extended conformal transformations. In particular, their features can be related to the perturbation…