PSI - Issue 43

Jiří Tomáš / Procedia Structural Integrity 43 (2023) 65– 70

70

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J. Toma´sˇ / Structural Integrity Procedia 00 (2023) 000–000

3. Higher-order and Weil Grassmanians Definition 1. (a) We define the equivalence ρ x , M on reg J r 0 ( R k , M ) For M = R m and x = 0 ∈ R m we simplify the notation to ρ and V . (b) We define the equivalence ( ρ A ) x , M on reg T A x M by ( j A ϕ x , j A ψ some j r x h ∈ Di ff r , M x . Further, we define the partition V x , M on reg J r 0 ( R k , M ) x by ( j r

0 ψ x = j r

r 0 ϕ x for

r 0 ψ x ) ∈ ρ x , M i ff j r

x h ◦ j

0 ϕ x , j

x formed by decomposition classes of ρ x , M .

x ) ∈ ( ρ A ) x , M i ff there are j r

0 ϕ x , 0 ∈ j x M consisting of A ϕ x ,

j r 0 ψ x , 0 ∈ j 0 ψ x , 0 ) ∈ ρ x , M . Further, we define the partition ( V A ) x , M on reg T A m and x = 0 we simplify the notation to ρ A and V A . Proposition 1. There is the unique structure of a smooth manifold on the space of right cosets G r k , m \ G r the decomposition classes of ( ρ A ) x , M . For M = R A ψ x such that ( j r 0 ϕ x , 0 , j r

k for which the

projection ˆ p : G r structure group G r

r k , m \ G

r k forms a surjective submersion. Moreover, ˆ p determines the principal bundle with the

k → G

k , m acting in the free and transitive way on fibres of ˆ p.

r , M k , m ) ( x , 0 ) \ inv J r

For any x ∈ M, the projection ˆ p x , M : inv J r 0 ( R

k , M × R k − m )

k , M × R k − m ) k , m ) ( x , 0) \ inv J r

0 ( R

( x , 0 ) defined

( x , 0 ) → (Di ff

by j r 1 ◦ j r 0 η x ] ρ determines the unique smooth manifold structure on (Di ff r , M R k − m ) ( x , 0) with respect to which ˆ p x , M is a surjective submersion. Finally, there is a a principal bundle structure on ˆ p x , M with the structure group (Di ff r , M k , m ) ( x , 0) acting in the free and transitive way on fiebers from the left. Identifying the right cosets from Proposition 1 to V -classes from Definition 1 we obtain Corollary 1. There is a principal bundle structure on ˆ p # : inv J r 0 ( R k , R m ) 0 → V with the structure group G r m G r m × { j r 0 id R k − m } acting from the left. This is the reduction of the principle bundle ˆ p : G r k → G r k , m \ G r k from Proposition 1. Analogously, there is a principal bundle structure on ˆ p # x , M : inv J r 0 ( R k , M ) x → V x , M with the structure group Di ff r x , M Di ff r x , M ×{ j r 0 id R k − m } obtained as a reduction of the principal bundle ˆ p x , M from Proposition 1. Definition 2. The basis V of ˆ p # is said to be the r-th order Grassmannian Gr ( r , k , m ) and the basis V x , M of ˆ p # x , M the r-th order Grassmannian Gr x ( r , k , M ) at x ∈ M. We define the bundle functor Gr ( r , k , M ) on m-dimensional manifolds and their local di ff eomorphisms by Gr ( r , k , M ) = x ∈ M Gr x ( r , k , M ) and Gr ( r , k , f ) by [ j r 0 ϕ ] ρ x , M → [ j r 0 ( f ◦ ϕ ] ρ f ( x ) , N for f : M → N, which can be identified with the bundle functor P r M [Gr ( r , k , m ) , ] in the sense of Kola´rˇ et al (1993) with the left action of G r m on the standard fiber Gr ( r , k , m ) defined by the composition of jets. We generalize the higher-order Grassmannian from A = D r k to the general Weil algebra A = D r k / I as follows. Proposition 2. There is a unique manifold structure on V A with respect to which the factorization map ˆ p # A : reg T A 0 R m → V A with respect to ρ A from Definition 1 is a surjective submersion. The same holds for ( ˆ p # A ) x , M : reg T A x M → ( V A ) x , M in case of the more general situation concerning M and x ∈ M. Definition 3. V A = Gr ( A , m ) with the manifold structure above is said to be the Weil Grassmannian. Proposition 3. There is a principal bundle structure on p A , R m ◦ ˆ p # : reg J r 0 ( R k , R m ) 0 → Gr ( A , m ) with the structure group G r m , which acts in the free and transitive way on fibers from the left. Fibers are formed by the elements T A 0 h T r 0 h = j r 0 h (Toma´sˇ (2017)) and are still the same. Extending Gr ( A , m ) to Gr ( A , M ) x = ( V A ) x , M and Gr ( A , M ) = x ∈ M Gr ( A , M ) x we obtain the bundle functor P r M [Gr ( A , m ) , ] on Gr ( A , M ) with of G r m from Definition 2 modified to Gr ( A , m ) , applying the coincidence of T A -morphism with r-jets from Toma´sˇ (2017). References Epstein, M., Elzanowski, M., 2007. Material Inhomogeneities and their Evolution. Springer, Berlin. Ehresmann, C., 1954. Extension du Calcul des jets aux jets non-holonomes. C. R. Acad. Sci 239, 1763-1764. Kainz, G., Michor, P. W., 1987. Natural transformations in di ff erential geometry. Czech. Math. J. 37, 584-607. Kola´ˇr, I., Michor, P. W., and Slova´k, J., 1993. Natural Operations in Di ff erential Geometry. Springer, Berlin. Kola´ˇr, I., 1986. Covariant approach to natural transformations of Weil bundles. Comment. Math. Univ. Carolinae, 99-105. Kuresˇ, M., 2014. Weil algebras associated to functors of third order semiholonomic velocities. Math. Journal of Okayama Univ. 56, 117-127. Sharpe, R. W., 1997. Di ff erential Geometry. Springer, Berlin. Toma´sˇ, J., 2017. A general rigidity result on A -covelocities and A -jets. Czech. Math. J. 67, 297-319. Wang, C. C., On the geometric structure of simple bodies. Arch. Rat. Mech. Anal. 27, 33-94. Arch. Rat. Mech. Anal. 27, 33-94. Wang, C. C., Truedell, C., Introduction to Rational Elasticity. Nordho ff , Leyden. 0 ( R k , M × 0 η x → j r 0 ¯ α x ◦ [( j r 0 ¯ α x ) −