Curvature, second variation and Jacobi fields in elastic diffeology
Abstract
We develop curvature theory on Blohmann's elastic diffeological spaces. We introduce curvature-tensorial connections, whose curvature factors uniquely through T3M, and show that this property holds on finite-dimensional manifolds and is preserved on mapping spaces. Using a plotwise commutator identity and the elasticity axioms, we establish the classical curvature symmetries for curvature-tensorial Levi-Civita connections. We define sectional curvature and a curvature form on the algebraic exterior square, and demonstrate an obstruction to the usual Ricci trace construction: on C∞(S1,S2), the relevant curvature endomorphism extends boundedly to the L2 completion but is not trace class. We express the second variation of the energy by the index form and characterize the null space of the index form by Jacobi fields. Given a geodesic flow, we construct Jacobi fields with prescribed initial data along its flow lines; they are unique when the real line is a curve object for the space. Finally, non-positive sectional curvature excludes conjugate points, while a negative index direction obstructs length minimization; the latter result does not require the curve-object property and applies to mapping spaces such as free loop spaces of complete Riemannian manifolds.
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BibTeX
@misc{shiobara2026curvature,
title = {{Curvature, second variation and Jacobi fields in elastic diffeology}},
author = {Yusuke Shiobara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29360},
eprint = {2610.29360},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29360},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Curvature, second variation and Jacobi fields in elastic diffeology
AU - Yusuke Shiobara
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29360
UR - https://arxiv.org/abs/2610.29360
AB - We develop curvature theory on Blohmann's elastic diffeological spaces. We introduce curvature-tensorial connections, whose curvature factors uniquely through T_3M, and show that this property holds on finite-dimensional manifolds and is preserved on mapping spaces. Using a plotwise commutator identity and the elasticity axioms, we establish the classical curvature symmetries for curvature-tensorial Levi-Civita connections. We define sectional curvature and a curvature form on the algebraic exterior square, and demonstrate an obstruction to the usual Ricci trace construction: on C^∞(S^1,S^2), the relevant curvature endomorphism extends boundedly to the L^2 completion but is not trace class. We express the second variation of the energy by the index form and characterize the null space of the index form by Jacobi fields. Given a geodesic flow, we construct Jacobi fields with prescribed initial data along its flow lines; they are unique when the real line is a curve object for the space. Finally, non-positive sectional curvature excludes conjugate points, while a negative index direction obstructs length minimization; the latter result does not require the curve-object property and applies to mapping spaces such as free loop spaces of complete Riemannian manifolds.
ER -