The almost Einstein operator for $(2, 3, 5)$ distributions

Sagerschnig, Katja; Willse, Travis

doi:10.5817/am2017-5-347

Cited by 11 publications

(21 citation statements)

References 26 publications

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“…as |t| → 0, uniformly in x. Using (49), (38), (33), and 2, where n has to be replaced by the homogeneous dimension n + r of M × R, we obtaiñ…”

Section: Lemma 3 the Differential Operator A+ ∂ ∂T Is A Rockland Opermentioning

confidence: 99%

“…x (A), a distribution on T x M which is supported at the unit element o x . The homogeneity formulated in (56) follows from (49), see also (33).…”

Section: Lemma 4 Consider the Heisenberg Principal Symbol σ R X (A) Amentioning

confidence: 99%

“…Recall that H is said to be of Cartan type [8] if it is bracket generating with growth vector (2,3,5). More precisely, H is of Cartan type if every point in M admits an open neighborhood U and there exist sections X , Y Rank-two distributions of Cartan type are also known as generic rank two distributions in dimension five, see [10,31,32,48,49], the condition on H being open with respect to the C 2 -topology. Their history can be traced back to Cartan's celebrated "five variables paper" [14].…”

Section: Generic Rank-two Distributions In Dimension Fivementioning

confidence: 99%

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The Heat Asymptotics on Filtered Manifolds

Dave

Haller

2019

J Geom Anal

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The short-time heat kernel expansion of elliptic operators provides a link between local and global features of classical geometries. For many geometric structures related to (non-)involutive distributions, the natural differential operators tend to be Rockland, hence hypoelliptic. In this paper, we establish a universal heat kernel expansion for formally self-adjoint non-negative Rockland differential operators on general closed filtered manifolds. The main ingredient is the analysis of parametrices in a recently constructed calculus adapted to these geometric structures. The heat expansion implies that the new calculus, a more general version of the Heisenberg calculus, also has a noncommutative residue. Many of the well-known implications of the heat expansion such as, the structure of the complex powers, the heat trace asymptotics, the continuation of the zeta function, as well as Weyl's law for the eigenvalue asymptotics, can be adapted to this calculus. Other consequences include a McKean-Singer type formula for the index of Rockland differential operators. We illustrate some of these results by providing a more explicit description of Weyl's law for Rumin-Seshadri operators associated with curved BGG sequences over 5-manifolds equipped with a rank-two distribution of Cartan type.

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“…as |t| → 0, uniformly in x. Using (49), (38), (33), and 2, where n has to be replaced by the homogeneous dimension n + r of M × R, we obtaiñ…”

Section: Lemma 3 the Differential Operator A+ ∂ ∂T Is A Rockland Opermentioning

confidence: 99%

“…x (A), a distribution on T x M which is supported at the unit element o x . The homogeneity formulated in (56) follows from (49), see also (33).…”

Section: Lemma 4 Consider the Heisenberg Principal Symbol σ R X (A) Amentioning

confidence: 99%

Section: Generic Rank-two Distributions In Dimension Fivementioning

confidence: 99%

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The Heat Asymptotics on Filtered Manifolds

Dave

Haller

2019

J Geom Anal

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“…3, we obtain a large class of explicit examples of twistor distributions for each special real algebraic type of the Cartan quartic. Additionally, one obtains that the associated (3,2) signature conformal structure on an open subset of N + has an Einstein representative whose conformal holonomy is a subgroup of SL 3 (R) ⊂ G * 2 ⊂ SO 4,3 ; this was also observed in [31]. The main purpose of Sect.…”

Section: Introduction and Main Resultsmentioning

confidence: 58%

Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

2022

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We study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type D, we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

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“…6). For completeness note that there exists a second well-known representation of G N C 2 as the symmetry group of a ball rolling on a larger fixed ball without slipping or twisting, when the ratio of the balls radii is 1/3 [41][42][43][44][45]. Understanding the exceptional Lie groups as the symmetry groups of naturally occurring objects is a long-standing program in mathematics.…”

Section: Transformations and Automorphismsmentioning

confidence: 99%

Geometrical Applications of Split Octonions

Gogberashvili

Sakhelashvili

2015

Advances in Mathematical Physics

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It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties from conventional (3+1)-theory (e.g. number of dimensions, existence of maximal velocities, Heisenberg uncertainty, particle generations, etc.). This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonions, the noncompact form of the exceptional Lie group G2. This group generates specific rotations of (3+4)-vector parts of split octonions with three extra time-like coordinates and in infinitesimal limit imitate standard Poincare transformations. In this picture translations are represented by non-compact Lorentz-type rotations towards the extra time-like coordinates. It is shown how the G2 algebra's chirality yields an intrinsic left-right asymmetry of a certain 3-vector (spin), as well as a parity violating effect on light emitted by a moving quantum system. Elementary particles are connected with the special elements of the algebra which nullify octonionic intervals. Then the zero-norm conditions lead to free particle Lagrangians, which allow virtual trajectories also and exhibit the appearance of spatial horizons governing by mass parameters.

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The almost Einstein operator for $(2, 3, 5)$ distributions

Cited by 11 publications

References 26 publications

The Heat Asymptotics on Filtered Manifolds

The Heat Asymptotics on Filtered Manifolds

Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

Geometrical Applications of Split Octonions

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