We define and study noncommutative generalizations of submanifolds and quotient manifolds using the derivation‐based differential calculus introduced by M. Dubois‐Violette and P. Michor. We give examples to illustrate these definitions.

1.
M.
Dubois-Violette
, “
Dérivations et calcul différentiel non commutatif
,”
C. R. Acad. Sci. Paris
,
307
, Sér.
I
,
403
408
(
1988
).
2.
M.
Dubois-Violette
and
P. W.
Michor
, “
Dérivations et calcul différentiel non commutatif II
,”
C. R. Acad. Sci. Paris
319
, Sér.
I
,
927
931
(
1994
).
3.
M. Dubois-Violette and P. W. Michor, “Connections on central bimodules in noncommutative differential geometry,” Preprint LPTHE-Orsay 94/100, ESI preprint 210, q-alg/9503020, to appear in J. Geom. Phys.
4.
A.
Connes
, “
Non-commutative differential geometry
,”
Pub. IHES
62
,
257
(
1986
).
5.
M. Dubois-Violette, “Non-commutative differential geometry, quantum mechanics and gauge theory,” in Differential Geometric Methods in Theoretical Physics, Proceedings Rapallo, 1990, edited by C. Bartocci, U. Bruzzo, and R. Cianci, Lecture Notes in Physics, Vol. 375 (Springer-Verlag, New York, 1991).
6.
M. Gerstenhaber and S. D. Schack, “Algebraic cohomology and deformation theory,” in Deformation Theory of Algebras and Structures and Applications, edited by M. Hazewinkel and M. Gerstenhaber (Kluwer, Dordrecht, 1988), pp. 11–264.
7.
M. Gerstenhaber, A. Giaquinto, and S. D. Schack, “Quantum symmetry,” Quantum Groups, Proceedings Leningrad, 1990, edited by P. P. Kulish, Lecture Notes in Mathematics, Vol. 1510 (Springer-Verlag, New York, 1992).
8.
M. Schlichenmaier, “Some concepts of modern algebraic geometry: point, ideal and homomorphism,” gk-mp-9403/3.
9.
O. E. Lanford, “Selected topics in functional analysis,” in Statistical Mechanics and Quantum Field Theory, Les Houches 1970, edited by C. DeWitt and R. Stora (Gordon and Breach, New York, 1971).
10.
M.
Dubois-Violette
,
R.
Kerner
, and
J.
Madore
, “
Noncommutative differential geometry and new models of gauge theory
,”
J. Math. Phys.
31
,
323
(
1990
).
11.
A.
Cap
,
A.
Kriegl
,
P.
Michor
, and
J.
Vanzăura
, “
The Frölicher-Nijenhuis Bracket in Non Commutative Differential Geometry
,”
Acta Math. Univ. Comenianiae
62
,
17
49
(
1993
).
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