Riemannian isometric immersion

Definition

Let $(M,g_M)$ and $(N,g_N)$ be Riemannian manifolds. A Riemannian isometric immersion is a smooth map $f:M\to N$ such that

f^*g_N=g_M.

Pointwise, this means

g_N(df_pu,df_pv)=g_M(u,v)

for all $p\in M$ and $u,v\in T_pM$ . Positive-definiteness implies that $df_p$ is injective, so the pullback equation already forces $f$ to be a smooth immersion.

The adjective “isometric” here refers to the Riemannian tensors. It does not by itself assert that $f$ is injective as a map, an embedding, or distance-preserving for the global geodesic distance between arbitrary points.

Special cases

If $f$ is also a smooth embedding, it is an isometric embedding. If it is a diffeomorphism, it is a Riemannian isometry, and the pullback equation implies that its inverse is also an isometry. Thus isometric immersions compose, while Riemannian isometries are the isomorphisms among them.

The inclusion of a Riemannian submanifold equipped with the induced metric is an isometric immersion. A covering map equipped with the pulled-back metric is another example that need not be injective.

Added structures

For Hermitian or Kähler manifolds, a Riemannian isometric immersion need not preserve the complex structures. Adding the equation

df\circ J_M=J_N\circ df

gives a holomorphic isometric immersion, which preserves both the metrics and the associated fundamental two-forms.

References

John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018. DOI record. Relevant: Riemannian submanifolds, isometric immersions, and isometries.
Manfredo P. do Carmo, Riemannian Geometry, Birkhäuser, 1992. DOI record. Relevant: isometric immersions and induced metrics.