nLab Riemannian immersion




A Riemannian immersion or isometric immersion of Riemannian manifolds is an immersion ΣX\Sigma \hookrightarrow X of their underlying smooth manifolds which is also an isometry with respect to their Riemannian metrics.

Similarly, an isometric embedding is an isometric immersion which is also an embedding of smooth manifolds, hence of the underlying topological spaces.

The isometric immersion/embedding problem is to find isometric immersions/embeddings of Riemannian manifolds into large-dimensional but flat (Euclidean) spaces (e.g. Han & Hong 2006, Han & Lewicka 2023).


For the following purpose:


A (local) co-frame field on a smooth manifold XX is

  1. an open cover X^X\widehat{X} \twoheadrightarrow X

  2. differential 1-formsE=(E a) a=1 dim(X)Ω dR 1(X^)E = (E^a)_{a=1}^{dim(X)} \,\in\, \Omega^1_{dR}(\widehat{X})

such that on double intersections

X^× XX^pr 2pr 1X \widehat{X} \!\times_X\! \widehat{X} \underoverset{pr_2}{pr_1}{\rightrightarrows} X

the induced metric tensors agree:

δ abpr 1 *E apr 1 *E b=δ abpr 2 *E apr 2 *E b \delta_{a b} \, pr_1^\ast E^a \otimes pr_1^\ast E^b \;\; = \;\; \delta_{a b} \, pr_2^\ast E^a \otimes pr_2^\ast E^b

(where we use the Einstein summation convention, throughout).

A pair of such local co-frames is regarded as equivalent if their induced metric tensors agree on the common refinement of their respective open covers.

Adapted Darboux (co-)frames

Given a Riemannian manifold (X,g)(X,g) of dimension DD, then

  • an orthonormal local frame is an open cover p:X^Xp \colon \widehat{X} \twoheadrightarrow X equipped with a DD-tuple of vector fields

    V: DTX^ V \;\colon\; \mathbb{R}^{D} \xrightarrow{\;\;} T \widehat X

    such that at each point x^X^\widehat x \,\in\, \widehat{X} we have that

    g(V,V)(x) g\big(V,V\big)(x)

    is the canonical inner product on D\mathbb{R}^D, hence in components

    g(V a,V b)(x)=δ ab; g\big(V_a, V_b\big)(x) \;=\; \delta_{a b} \,;
  • an orthonormal local co-frame is an open cover p:X^Xp \colon \widehat{X} \twoheadrightarrow X equipped with a D\mathbb{R}^D-valued differential 1-form

    E:TX^ D E \;\colon\; T\widehat{X} \xrightarrow{\;\;} \mathbb{R}^D

    such that at each point x^X^\widehat x \,\in\, \widehat{X} we have

    δ abE aE b=g. \delta_{a b} \, E^a \otimes E^b \;=\; g \,.

Now given moreover an immersion ϕ:ΣX\phi \colon \Sigma \hookrightarrow X into a Riemannian manifold


The Darboux co-frame property (2) immediately implies that the pullback of the remaining frame components

e(e aϕ *E a) adim(Σ) e \;\coloneqq\; \big( e^a \;\coloneqq\; \phi^\ast E^a \big)_{ a \leq dim(\Sigma) }

constitute a local frame on Σ\Sigma. We may summarize this by saying that a local Darboux co-frame EE gives

(3)ϕ *E a={e a foradim(Σ) 0 foradim(Σ). \phi^\ast E^a \;=\; \left\{ \begin{array}{ll} e^a & \text{for}\; a \leq dim(\Sigma) \\ 0 & \text{for}\; a \geq dim(\Sigma) \,. \end{array} \right.

(cf. Griffiths & Harris 1979 (1.13)).

Incidentally, this situation (3) of Darboux co-frames, when lifted/applied to the bosonic coframe components of super spacetimes, has later come to be known as the “embedding condition” in the “super-embedding approach” to super p-branes [Sorokin 2000 (4.36-37); Bandos 2011 (2.6-2.9); Bandos & Sorokin 2023 (5.13-14)], strenghtening the original “geometrodynamical condition” of Bandos et al. 1995 (2.23), which is just the first condition in (3).


Given an immersion into a Riemannian manifold, local Darboux (co-)frames always exist.


Given an immersion ι:ΣX\iota \colon \Sigma \to X, consider any point σΣ\sigma \in \Sigma. Since the immersion is locally an embedding (see here), there exists an open neighbourhood σUΣ \sigma \in U \subset \Sigma such that ϕ |U:UX\phi_{\vert U} \colon U \to X is the embedding of a submanifold. Therefore (by this Prop.) there exists an open neighbourhood UXU' \subset X of ι(σ)X\iota(\sigma) \in X which serves as a coordinate chart XUϕ nX \supset U' \xrightarrow{\phi} \mathbb{R}^n for XX and a slice chart for ΣX\Sigma \subset X in that it exhibits ΣU\Sigma \cap U' as a rectilinear hyperplane in ϕ(U) n\phi(U') \subset \mathbb{R}^n.

Hence in this slice coordinate chart a local frame for XX is given by the nn canonical coordinate vector fields, with the first kk of them forming a local frame field on Σ\Sigma

{ 1,, k}{ 1,, k, k+1,, n}. \big\{ \partial_1, \cdots, \partial_{k} \big\} \hookrightarrow \big\{ \partial_1, \cdots, \partial_{k}, \, \partial_{k+1}, \cdots, \partial_n \big\} \,.

From this local frame the Gram-Schmidt process produces an orthonormal frame, first for Σ\Sigma and then extended to XX:

{v 1,,v k}{v 1,,v k,v k+1,,v n},g(v a,v b)=δ ab \big\{ v_1, \cdots, v_{k} \big\} \hookrightarrow \big\{ v_1, \cdots, v_{k}, \, v_{k+1}, \cdots, v_n \big\} \,, \;\;\; g(v_a, v_b) = \delta_{a b} \,

This demonstrates the existence of orthonormal local frame fields (cf. Kayban 2021, Prop. 3.1).

To obtain the desired orthonormal local co-frame field we just dualize this local frame field:

By construction, the matrix (v a μ) a,μ\big(v_a^\mu\big)_{a,\mu} of components of the above frame (given by v a μ μ=v av_a^\mu \partial_\mu = v_a) is block diagonal (the upper diagonal block being the local frame on Σ\Sigma).

This means (cf. e.g. here) that also the inverse matrix

(e μ a) μ,a((v a μ) μ,a) 1 \big(e^a_\mu\big)_{\mu,a} \,\coloneqq\, \Big(\big(v_a^\mu\big)_{\mu,a}\Big)^{-1}

is block diagonal, with its upper diagonal block being the inverse matrix of the original upper left block.

This gives the desired coframe field:

e ae μ adx μ e^{a} \,\coloneqq\, e^a_\mu \, \mathrm{d}x^\mu

which is orthonormal on XX

e ae a=g(,) e^a \otimes e_a \;=\; g(-,-)


v a μg μνv b ν=δ abg μν=e μ aδ abe ν b. v_a^\mu g_{\mu \nu} v_b^\nu \;=\; \delta_{a b} \;\;\;\; \Leftrightarrow \;\;\;\; g_{\mu \nu} \;=\; e^a_\mu \delta_{a b} e^b_\nu \,.

and which is Darboux by the block-diagonal structure of (e μ a)\big(e^a_\mu\big).

(For the further generality of sequences of Darboux frames for suitable sequences of immersions, see Giron 2020 and Chen & Giron 2021, Thm. 2.2.)


In particular, an immersion ι:ΣX\iota \colon \Sigma \hookrightarrow X of Riemannian manifolds is isometric iff around each point ι(σ)\iota(\sigma) any of its Darboux coframe fields pull back to locally induce the given metric on Σ\Sigma.

Second fundamental form

Given XX a Riemannian manifold and ϕ:ΣX\phi \colon \Sigma \to X an immersion, choose a Darboux co-frame field E(E a) a=1 dim(X)E \equiv (E^a)_{a =1}^{dim(X)} (which exists by Prop. ), hence so that

ϕ *E a=0 fora{dim(Σ)+1,,dim(X)} (e aϕ *E a) a=1 dim(Σ) is a co-frame onΣ \begin{array}{l} \phi^\ast E^a \;=\; 0 & \text{for} \; a \in \big\{dim(\Sigma)+1, \cdots, dim(X)\big\} \\ \big( e^a \;\coloneqq\; \phi^\ast E^a \big)_{a = 1}^{dim(\Sigma)} & \text{is a co-frame on}\; \Sigma \end{array}

For brevity we will say that a{1,,dim(X)}a \in \{1, \cdots, dim(X)\} is

  • tangential if a{1,,dim(Σ)}a \in \big\{1, \cdots, dim(\Sigma)\big\}

  • transversal if a{dim(Σ)+1,,dim(X)}a \in \big\{dim(\Sigma)+1, \cdots, dim(X)\big\}.

Now let Ω\Omega be the unique torsion-free connection for EE, in that

(4)dE aΩ a bE b=0. \mathrm{d} E^a - \Omega^a{}_b E^b \;=\; 0 \,.

and denote the pullback of its tangential and transversal components, respectively, by

ω a b ϕ *Ω a b for tangentialand tangentialb b 1b 2 ae b 1 ϕ *Ω a b 2 for transversalaand tangentialb 2. \begin{array}{cccl} \omega^a{}_b &\coloneqq& \phi^\ast \Omega^a{}_b & \text{for tangential}\;\text{and tangential}\;b \\ {Ⅱ}^a_{b_1 b_2} e^{b_1} &\coloneqq& \phi^\ast \Omega^a{}_{b_2} & \text{for transversal}\;a\;\text{and tangential}\;b_2 \,. \end{array}

Then the pullback of the torsion constraint (4) is equivalent to thid pair of conditions on Σ\Sigma:

{de aω a be b=0 for tangentiala b 1b 2 ae b 1e b 2=0 for transversala \left\{ \begin{array}{ll} \mathrm{d} e^a - \omega^a{}_b \, e^b \;=\; 0 & \text{for tangential}\;a \\ {Ⅱ}^a_{b_1 b_2} e^{b_1} e^{b_2} \;=\; 0 & \text{for transversal}\;a \end{array} \right.

The first one just says that ω\omega is the torsion-free connection with respect to the induced coframe ee on Σ\Sigma.

The second one says that the skew symmetric part [b 1b 2] a=0{Ⅱ}^a_{[b_1 b_2]} = 0 vanishes, hence that

b 1b 2 a= b 2b 1 a {Ⅱ}^a_{b_1 b_2} \;=\; {Ⅱ}^a_{b_2 b_1}

is symmetric in its tangential indices b ib_i. This symmetric tensor on Σ\Sigma is called the second fundamental form of the immersion ϕ\phi.

(e.g. Berger, Bryant & Griffiths 1983, p. 819; Chavel 1993 (II.2.12); Wang 2024, Def. 2.2; for alternative discussion not using co-frames see also Chavel 1993 §II.2; Lee 2018, pp. 225)


Last revised on May 22, 2024 at 19:56:26. See the history of this page for a list of all contributions to it.