arXiv subjects

Tohru Morimoto

Publications and source records attributed to Tohru Morimoto.

4 recordsLinked to original sources

arXiv2024

Prolongations, invariants, and fundamental identities of geometric structures

Working in the framework of nilpotent geometry, we give a unified scheme for the equivalence problem of geometric structures which extends and integrates the earlier works by Cartan, Singer-Sternberg, Tanaka, and Morimoto. By giving a new formulation of the higher order geometric structures and the universal frame bundles, we reconstruct the step prolongation of Singer-Sternberg and Tanaka. We then investigate the structure function $γ$ of the complete step prolongation of a normal geometric structure by expanding it into components $γ= κ+ τ+ σ$ and establish the fundamental identities for $κ$, $τ$, $σ$. This then enables us to study the equivalence problem of geometric structures in full generality and to extend applications largely to the geometric structures which have not necessarily Cartan connections. Among all we give an algorithm to construct a complete system of invariants for any higher order normal geometric structure of constant symbol by making use of generalized Spencer cohomology group associated to the symbol of the geometric structure. We then discuss thoroughly the equivalence problem for geometric structure in both cases of infinite and finite type. We also give a characterization of the Cartan connections by means of the structure function $τ$ and make clear where the Cartan connections are placed in the perspective of the step prolongations.

math.DG↗

arXiv2023

Extrinsic geometry and linear differential equations of $\mathfrak{sl}_3$-type

As an application of the general theory on extrinsic geometry, we investigate extrinsic geometry in frag varieties and systems of linear PDE's for a class of special interest associated with the adjoint representation of $\mathfrak{sl}(3)$. We carry out a complete local classification of the homogeneous structures in this class. As a result, we find 7 kinds of new systems of linear PDE's of second order on a 3-dimensional contact manifold each of which has a solution space of dimension 8. Among them there are included a system of PDE's called contact Cayley's surface and one which has $\mathfrak{sl}(2)$ symmetry.

math.DG↗

arXiv2021

Extrinsic Geometry and Linear Differential Equations

We give a unified method for the general equivalence problem of extrinsic geometry, on the basis of our formulation of a general extrinsic geometry as that of an osculating map $φ\colon (M,\mathfrak f) \to L/L^0 \subset \operatorname{Flag}(V,ϕ)$ from a filtered manifold $(M,\mathfrak f)$ to a homogeneous space $L/L^0$ in a flag variety $\operatorname{Flag}(V,ϕ)$, where $L$ is a finite-dimensional Lie group and $L^0$ its closed subgroup. We establish an algorithm to obtain the complete systems of invariants for the osculating maps which satisfy the reasonable regularity condition of constant symbol of type $(\mathfrak g_-, \operatorname{gr} V, L)$. We show the categorical isomorphism between the extrinsic geometries in flag varieties and the (weighted) involutive systems of linear differential equations of finite type. Therefore we also obtain a complete system of invariants for a general involutive systems of linear differential equations of finite type and of constant symbol. The invariants of an osculating map (or an involutive system of linear differential equations) are proved to be controlled by the cohomology group $H^1_+(\mathfrak g_-, \mathfrak l / \bar{\mathfrak g})$ which is defined algebraically from the symbol of the osculating map (resp. involutive system), and which, in many cases (in particular, if the symbol is associated with a simple Lie algebra and its irreducible representation), can be computed by the algebraic harmonic theory, and the vanishing of which gives rigidity theorems in various concrete geometries. We also extend the theory to the case when $L$ is infinite dimensional.

math.DG↗

arXiv2017

Lie algebras attached to Clifford modules and simple graded Lie algebras

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo $H$-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type $B_n$ with $|2|$-grading do not contain non-Heisenberg pseudo $H$-type Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types $A_n$, $C_n$ and $D_n$ provide such a possibility. Among exceptional algebras only $F_4$ and $E_6$ contain non-Heisenberg pseudo $H$-type Lie algebras as their negative part of $|2|$-grading. An analogous question addressed to real simple graded Lie algebras is more difficult, and we give results revealing the main differences with the complex situation.

math.DG↗