TY - JOUR
T1 - Riemannian metrics on differentiable stacks
AU - del Hoyo, Matias
AU - Fernandes, Rui Loja
N1 - Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2019/6/1
Y1 - 2019/6/1
N2 - We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.
AB - We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.
UR - http://www.scopus.com/inward/record.url?scp=85055949922&partnerID=8YFLogxK
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U2 - 10.1007/s00209-018-2154-6
DO - 10.1007/s00209-018-2154-6
M3 - Article
AN - SCOPUS:85055949922
SN - 0025-5874
VL - 292
SP - 103
EP - 132
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 1-2
ER -