We study infinitesimal Einstein deformations on compact flat manifolds and on product manifolds. Moreover, we prove refinements of results by Koiso and Bourguignon which yield obstructions on the existence of infinitesimal Einstein deformations under certain curvature conditions.
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48 results for “Infinitesimal Einstein deformations”
We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations pr…
Study on complex Grassmannians' rigidity using Einstein deformations.
problem Characterizing integrable infinitesimal Einstein deformations of complex Grassmannians.
method Analyzing the integrability to second order of infinitesimal deformations using Koiso's obstruction polynomial.
result Characterized integrable deformations as an explicit variety in su(n), showing is isolated for odd n.
Study on deformations of Einstein and nearly G2 structures in 3-Sasaki manifolds.
problem Deformation theory of Einstein and nearly G2 structures in 3-Sasaki manifolds.
method Systematic study of deformation theory, focusing on infinitesimal deformations and their parametrization via eigenfunctions of the basic Laplacian.
result Infinitesimal Einstein deformations of coincide with infinitesimal deformations of .
Study infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.
problem Infinitesimal deformations of Killing spinors on nearly parallel G2-manifolds.
method Examined using the correspondence between nearly parallel G2-structures and Killing spinors.
result Identified that the space of Rarita-Schwinger fields coincides with a subspace of the eigenspace of the Laplacian.
The paper studies Einstein metrics on specific manifolds and their rigidity properties.
problem Investigating rigidity of Einstein metrics on homogeneous Gray manifolds.
method Computing coindex and analyzing infinitesimal deformations of Einstein metrics.
result Infinitesimal Einstein deformations on are not integrable.
Starting with a compact hyperbolic cone-manifold of dimension n > 2, we study the deformations of the metric in order to get Einstein cone-manifolds. If the singular locus is a closed codimension 2 submanifold and all cone angles are smaller than 2 pi, we show that there is no non-trivial infinitesimal Einstein deforma…
It is well-known that every 6-dimensional strictly nearly Kähler manifold is Einstein with positive scalar curvature . Moreover, one can show that the space of co-closed primitive (1,1)-forms on is stable under the Laplace operator . Let denote the -eigenspace of the restriction o…
Study shows unique Einstein metrics on and related spaces.
problem Rigidity of Einstein metrics on and related spaces.
method Proof of rigidity using infinitesimal deformations and connections to Ricci flow.
result Bi-invariant Einstein metric on is isolated in the moduli space of Einstein metrics.
On the Rigidity of math.DG
Revisits Koiso's rigid metrics on complex projective spaces.
problem Computing obstructions to integrability of deformations.
method Elementary complex differential geometry.
result Computes Koiso's obstruction on .
Study second order integrability of Einstein deformations on Riemannian and Kähler manifolds.
problem Integrability condition for second order infinitesimal Einstein deformations.
method New expression for integrability condition, Koiso obstruction simplification.
result Complete description of integrable deformations on complex 2-plane Grassmannian.
Study on Einstein deformations of negative Kähler Einstein metrics.
problem Understanding Einstein deformations of Kähler Einstein metrics.
method Relate second order Einstein deformation theory to complex geometry, gauge normalise, and use Taylor expansion.
result Taylor expansion to order two of an Einstein deformation is determined by and the divergence of the Kodaira-Spencer bracket.
Sandwich operators and Einstein deformations of compact symmetric spaces related to Jordan algebrasmath.DG
Study on deformations of symmetric spaces using Jordan algebras.
problem Deformability of symmetric Einstein metrics on compact Lie algebras.
method Developed sandwich operators and quadratic Casimir operators for compact Lie algebras; calculated obstruction integrals from invariant polynomials; explored relation to simple Jordan algebras.
result Proved the nonlinear instability of most infinitesimally deformable irreducible compact symmetric spaces.
We study the infinitesimal deformations of a proper nearly parallel G_2-structure and prove that they are characterized by a certain first order differential equation. In particular we show that the space of infinitesimal deformations modulo the group of diffeomorphisms is isomorphic to a subspace of co-closed $Λ^3_{27…
It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional at compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.
Rigidity of Fubini-Study metric on odd complex Grassmannians.
problem Characterize infinitesimal deformations of the Fubini-Study metric on complex Grassmannians.
method Explicit description of infinitesimal Einstein deformations, integration analysis.
result Fubini-Study metric on odd complex Grassmannians is rigid.
The study examines stability of specific geometric flows.
problem Stability of Pluriclosed and Generalized Ricci solitons.
method Analyzes the second variation of generalized Einstein--Hilbert functional and infinitesimal deformations.
result Stability of the flows and solitons under specific conditions.
The paper is centered around a new proof of the infinitesimal rigidity of smooth closed surfaces with everywhere positive Gauss curvature. We use a reformulation that replaces deformation of an embedding by deformation of the metric inside the body bounded by the surface. The proof is obtained by studying derivatives o…
Starting with a compact hyperbolic cone-manifold of dimension greater than or equal to 3, we study the deformations of the metric with the aim of getting Einstein cone-manifolds. If the singular locus is a closed codimension 2 submanifold and all cone angles are smaller than 2 pi, we show that there is no non-trivial i…
Study on stability of quaternion-Kähler manifolds using eigenvalue estimates.
problem Stability problem for positive quaternion-Kähler manifolds.
method Description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and symmetric 2-tensors. Improved eigenvalue estimates for the Hodge-Laplacian on 2-forms.
result Sharp lower bound for the first non-zero eigenvalue on the parallel subbundle Sym^2 E of the 2-form bundle.
Study shows Einstein structures on 4-manifolds are rigid.
problem Rigidity of Einstein structures in four dimensions.
method Examined deformations of the round four-sphere and analyzed self-dual structure of Einstein manifolds.
result Any deviation from the standard metric of the round four-sphere breaks the Einstein condition.
A differential calculus, differential geometry and the E-R Gravity theory are studied on noncommutative spaces. Noncommutativity is formulated in the star product formalism. The basis for the gravity theory is the infinitesimal algebra of diffeomorphisms. Considering the corresponding Hopf algebra we find that the defo…
The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. This approach is in a sense dual to using derivatives of the volume i…
Study on Einstein deformations and desingularizations, finding some 4-orbifolds cannot be limits of smooth metrics.
problem Whether every Einstein 4-orbifold is a limit of smooth Einstein 4-manifolds.
method Analysis of integrability of deformations through variations of Schoen's Pohozaev identity and introduction of preserved integral quantities.
result Spherical and hyperbolic 4-orbifolds with the simplest singularities cannot be limits of smooth Einstein 4-manifolds.
The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidea…
Using spin structure we prove that Kähler-Einstein metrics with nonpositive scalar curvature are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Moreover if all infinitesimal complex deformation of the complex structure are integrable, th…
We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The class of noncommutative spaces studied is very rich. Non-anticommutative superspa…
Study of instantons on Stiefel manifold with and Sasakian structures.
problem Characterizing and classifying instantons on Stiefel manifold.
method Reductive decomposition, spinorial approach, Weitzenböck-type formula with torsion.
result Classification of invariant connections and rigidity results for -instantons.
We study infinitesimal conformal deformations of a triangulated surface in Euclidean space and investigate the change in its extrinsic geometry. A deformation of vertices is conformal if it preserves length cross-ratios. On one hand, conformal deformations generalize deformations preserving edge lengths. On the other h…
We study deformations of shrinking Ricci solitons on a compact manifold M, generalising the classical theory of deformations of Einstein metrics. Using appropriate notions of twisted slices S_f inside the space of all Riemannian metrics on M, we define the infinitesimal solitonic deformations and the local solitonic pr…
Study on deformations of holomorphic Cartan geometries, focusing on flat cases.
problem Deformation of holomorphic Cartan geometries on complex manifolds.
method Computing infinitesimal deformations and analyzing the forgetful map.
result The forgetful map from infinitesimal deformations of a flat holomorphic Cartan geometry to the underlying flat principal bundle is an isomorphism.
Study infinitesimal deformations of Lie algebroid pairs.
problem Infinitesimal deformations of Lie algebroid pairs.
method Investigate isomorphism classes of infinitesimal deformations of modulo automorphisms from exponentials of derivations of and those from the exponentials of inner derivations of .
result Find the associated governing -algebras in the sense of extended deformation theory.
In this paper, an obstruction against the integrability of certain infinitesimal solitonic deformations is given. Using this obstruction, we show that the complex projective spaces of even complex dimension are rigid as Ricci solitons although they have infinitesimal solitonic deformations.
Introducing the deformation theory of holomorphic Cartan geometries, we compute infinitesimal automorphisms and infinitesimal deformations. We also prove the existence of a semi-universal deformation of a holomorphic Cartan geometry.
New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.
problem Proving non-degeneracy of Poincaré-Einstein metrics.
method Proved non-degeneracy for 4D metrics satisfying a chiral curvature inequality.
result 4D Poincaré-Einstein metrics are non-degenerate if curvature is negative definite.
CR Killing operator derived from tractor calculus for CR structures.
problem Analyzing CR structures and their deformations.
method Tractor calculus and BGG operators applied to compatible almost CR structures.
result CR Killing operator is a first BGG operator for the modified adjoint tractor connection.
Let be a polyhedron. It was conjectured that if is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assu…
We describe infinitesimal deformations of constant mean curvature surfaces of finite type in the 3-sphere. We use Baker-Akhiezer functions to describe such deformations, as well as polynomial Killing fields and the corresponding spectral curve to distinguish between isospectral and non-isospectral deformations.
There are very few general theorems on the kernel of the well-known Lichnerowicz Laplacian. In the present article we consider the geometry of the kernel of this operator restricted to covariant (not necessarily symmetric or skew-symmetric) tensors. Our approach is based on the analytical method, due to Bochner, of pro…
Computes deformations of parabolic structures on Riemann surfaces.
problem Infinitesimal deformations of parabolic connections and opers.
method Computes infinitesimal deformations of quadruples (X, S, E*, D) and (X, S, D).
result Monodromy map is an immersion from the moduli space of triples to the character variety.
Develops deformation theory for symplectic foliations using -algebras.
problem Deformation of symplectic foliations.
method Uses -algebras to control deformation problems.
result Establishes a correspondence between small deformations and Maurer-Cartan elements of -algebra.
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
This paper applies the authors' forthcoming work, "Affine deformations of a three-holed sphere" in Lorentzian geometry to prove a result in hyperbolic geometry. Namely, an infinitesimal deformation of a hyperbolic structure of a three-holed sphere which infinitesimally lengthens the three boundary components infinitesi…
Study strip deformations of hyperbolic polygons with decorated vertices.
problem Understanding deformations of hyperbolic polygons with decorated vertices.
method Analyzing strip deformations of ideal hyperbolic polygons with horoballs.
result Arc complexes parameterize uniformly lengthening deformations.
New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
Classifies hypersurfaces preserving Gauss map with two conditions.
problem Classify nonflat hypersurfaces preserving the Gauss map.
method Analyzes conditions for hypersurfaces to preserve the Gauss map.
result Hypersurfaces must be Kaehler.
New proof shows perturbed non-compact Einstein spaces attract to unique global solution.
problem Proving global solutions for perturbed non-compact negative Einstein spaces.
method Developed energy estimates for a hyperbolic system of Maxwell type.
result Global unique solution for perturbed non-compact negative Einstein spaces.
Study on deformation theory of nearly G2 manifolds with obstructions.
problem Deformation theory of nearly G2 manifolds with obstructions.
method Study of real Killing spinors and cohomology of nearly G2 manifolds.
result Infinitesimal deformations of nearly G2 structures are obstructed in general.