Topological Obstructions to Riemannian-Lorentzian Continuation
Abstract
Motivated by problems in instanton continuation, we establish geometro-topological selection rules for continuing a Riemannian region into a Lorentzian region through a smooth transverse change of signature. Given a collar-gluing M=MR̅ ∪ℋML̅ of a smooth manifold M along a hypersurface ℋ, we show that an admissible signature-changing metric of the form g̃=g−fV♭⊗ V♭ with g a Riemannian metric, having Riemannian signature on MR and Lorentzian signature on ML, exists if and only if a relative Euler class obstruction on ML̅ vanishes. This obstruction combines ordinary Euler characteristics χ in a manner that depends on which components of ℋ represent big bangs and which represent big crunches. It also restricts the choice of MR; for example, if M is closed and dim M is even, then it requiresχ(MR̅)=χ(M). We also determine and classify the possible compactness types of MR̅, ℋ, and ML̅ occurring in such a collar-gluing construction, and construct explicit signature-changing metrics in each case. Topology change can be entirely contained within the Riemannian region when M is the interior of a compact manifold with boundary.
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BibTeX
@misc{rieger2026topological,
title = {{Topological Obstructions to Riemannian-Lorentzian Continuation}},
author = {Nathalie E. Rieger and Ödül Tetik},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27136},
eprint = {2609.27136},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27136},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Topological Obstructions to Riemannian-Lorentzian Continuation
AU - Nathalie E. Rieger
AU - Ödül Tetik
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.27136
UR - https://arxiv.org/abs/2609.27136
AB - Motivated by problems in instanton continuation, we establish geometro-topological selection rules for continuing a Riemannian region into a Lorentzian region through a smooth transverse change of signature. Given a collar-gluing M=M_R̅ ∪_ℋM_L̅ of a smooth manifold M along a hypersurface ℋ, we show that an admissible signature-changing metric of the form g̃=g−fV^♭⊗ V^♭ with g a Riemannian metric, having Riemannian signature on M_R and Lorentzian signature on M_L, exists if and only if a relative Euler class obstruction on M_L̅ vanishes. This obstruction combines ordinary Euler characteristics χ in a manner that depends on which components of ℋ represent big bangs and which represent big crunches. It also restricts the choice of M_R; for example, if M is closed and dim M is even, then it requiresχ(M_R̅)=χ(M). We also determine and classify the possible compactness types of M_R̅, ℋ, and M_L̅ occurring in such a collar-gluing construction, and construct explicit signature-changing metrics in each case. Topology change can be entirely contained within the Riemannian region when M is the interior of a compact manifold with boundary.
ER -