Let Ω be a bounded and connected open subset of Rn with a Lipschitz-continuous boundary Γ, the set Ω being locally on the same side of Γ, and let Θ:Ω‾→Rn and Φ:Ω‾→Rn be two smooth enough “deformations” of the set Ω‾. Then the classical Korn inequality asserts that, when Θ=id, there exists a constant c such that‖v‖H1(Ω)≤c(‖v‖L2(Ω)+‖∇v+∇vT‖L2(Ω)) for all v∈H1(Ω), where v:=(Φ−id):Ω‾→Rn denotes the corresponding “displacement” vector field, and where the symmetric tensor field ∇v+∇vT:Ω‾→Sn is nothing but the linear part with respect to v of the difference between the metric tensor fields ∇ΦT∇Φ and I that respectively correspond to the deformations Φ and Θ=id.Assume now that the identity mapping id is replaced by a more general orientation-preserving immersion Θ∈C1(Ω‾;Rn). We then show in particular that, given any 1<p<∞ and any q∈R such that max{1,p2}≤q≤p, there exists a constant C=C(p,q,Θ) such that‖Φ−Θ‖W1,p(Ω)≤C(‖Φ−Θ‖Lp(Ω)+‖∇ΦT∇Φ−∇ΘT∇Θ‖Lq(Ω)q/p) for all Φ∈W1,2q(Ω) that satisfy det∇Φ>0 almost everywhere in Ω. Such an inequality thus constitutes an instance of a “nonlinear Korn inequality”, in the sense that the symmetric tensor field ∇ΦT∇Φ−∇ΘT∇Θ:Ω‾→Sn appearing in its right-hand side is now the exact difference between the metric tensor fields corresponding to the deformations Φ and Θ.We also show that, like in the linear case, an analogous nonlinear Korn inequality holds, but without the norm ‖Φ−Θ‖Lp(Ω) in its right-hand side, if the difference Φ−Θ vanishes on a subset Γ0 of Γ with dΓ-meas Γ0>0.The key to providing such nonlinear Korn inequalities is a generalization of the landmark “geometric rigidity lemma in H1(Ω)” established in 2002 by G. Friesecke, R.D. James, and S. Müller, as later extended to W1,p(Ω) by S. Conti.