Related papers: On the frame bundle adapted to a submanifold

Let $(M,g)$ be a Riemannian manifold, $L(M)$ be its frame bundle, $O(M)$ its orthonormal frame bundle. For a distribution $D$ on $M$ we define a subbundle $L(D)\subset L(M)$ or $O(D)\subset O(M)$ in a natural way. This allows us to consider…

We study the geometry of a $G$-structure $P$ inside the oriented orthonormal frame bundle ${\rm SO}(M)$ over an oriented Riemannian manifold $M$. We assume that $G$ is connected and closed, so the quotient ${\rm SO}(n)/G$, where $n=\dim M$,…

Let $(M,g)$ be a Riemannian manifold, $L(M)$ its frame bundle. We construct new examples of Riemannian metrics on $L(M)$, which are obtained from Riemannian metrics on the tangent bundle $TM$. We compute the Levi--Civita connection and…

We investigate the geometry of a normal bundle equipped with a $(p,q)$-metric, i.e., Riemannian metric of Cheeger-Gromoll type, to the submanifold of a Riemannian manifold. We derive all natural object as the Levi-Civita connection,…

In the present work we construct a lift of a metric $g$ on a 2-dimensional oriented Riemannian manifold $M$ to a metric $\hat{g}$ on the total space $P$ of the orthonormal frame bundle of $M$. We call this lift the \textit {Wagner lift}.…

One of the known mathematical descriptions of singularities in General Relativity is the b-boundary, which is a way of attaching endpoints to inextendible endless curves in a spacetime. The b-boundary of a manifold M with connection is…

We show Riemannian geometry could be studied by identifying the tangent bundle of a Riemannian manifold $\mathcal{M}$ with a subbundle of the trivial bundle $\mathcal{M} \times \mathcal{E}$, obtained by embedding $\mathcal{M}$…

We introduce the concept of ODD ('$\mathbf{O}$rthogonally $\mathbf{D}$egenerating on a $\mathbf{D}$ivisor') Riemannian metrics on real analytic manifolds $M$. These semipositive symmetric $2$-tensors may degenerate on a finite collection of…

We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove…

For a Riemannian manifold $(N,g)$, we construct a scalar flat metric $G$ in the tangent bundle $TN$. It is locally conformally flat if and only if either, $N$ is a 2-dimensional manifold or, $(N,g)$ is a real space form. It is also shown…

We consider the orthonormal frame bundle F(M) of a Riemannian manifold M. A construction of Sasaki defines a canonical Riemannian metric on F(M). We prove that for two closed Riemannian n-manifolds M and N, the frame bundles F(M) and F(N)…

Let $\mathcal M= (M,\mathcal O_\mathcal M)$ be a smooth supermanifold with connection $\nabla$ and Batchelor model $\mathcal O_\mathcal M\cong\Gamma_{\Lambda E^\ast}$. From $(\mathcal M,\nabla)$ we construct a connection on the total space…

Given an $m$-dimensional closed connected Riemannian manifold $M$ smoothly isometrically immersed in an $n$-dimensional Riemannian manifold $N$, we estimate the diameter of $M$ in terms of its mean curvature field integral under some…

We discuss some specializations of the frames of flat orthonormal frame bundles over geometries of indefinite signature, and the resulting symmetries of families of embedded Riemannian or pseudo-Riemannian geometries. The specializations…

Given a submanifold $S \subset \mathbb R^n$ of codimension at least three, we construct an asymptotically Euclidean Riemannian metric on $\mathbb R^n$ with nonnegative scalar curvature for which the outermost apparent horizon is…

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel…

Let $\overline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$. Consider an affine connection $\nabla$ on $M$ for which the boundary is at infinity. Then $\nabla$ is projectively compact of order $\alpha$ if the…

A conformal structure on a manifold $M^n$ induces natural second order conformally invariant operators, called M\"obius and Laplace structures, acting on specific weight bundles of $M$, provided that $n\ge 3$. By extending the notions of…

Substatic Riemannian manifolds with minimal boundary arise naturally in General Relativity as spatial slices of static spacetimes satisfying the Null Energy Condition. Moreover, they constitute a vast generalization of nonnegative Ricci…

We consider a regular distribution $\mathcal{D}$ in a Riemannian manifold $(M,g)$. The Levi-Civita connection on $(M,g)$ together with the orthogonal projection allow to endow the space of sections of $\mathcal{D}$ with a natural covariant…