nLab Riemannian immersion

Idea

A Riemannian immersion or isometric immersion of Riemannian manifolds is an immersion $\Sigma \hookrightarrow X$ 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).

Properties

For the following purpose:

Definition

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

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

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

such that on double intersections

$$\widehat{X} \!\times_X\! \widehat{X} \underoverset{pr_2}{pr_1}{\rightrightarrows} X$$

the induced metric tensors agree:

$$\delta_{a b} \, pr_1^\ast E^a \otimes pr_1^\ast E^b \;\; = \;\; \delta_{a b} \, pr_2^\ast E^a \otimes pr_2^\ast E^b$$

(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 $(X,g)$ of dimension $D$, then

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

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

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

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

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

  $$g\big(V_a, V_b\big)(x) \;=\; \delta_{a b} \,;$$

• an orthonormal local co-frame is an open cover $p \colon \widehat{X} \twoheadrightarrow X$ equipped with a $\mathbb{R}^D$-valued differential 1-form

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

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

  $$\delta_{a b} \, E^a \otimes E^b \;=\; g \,.$$

Now given moreover an immersion $\phi \colon \Sigma \hookrightarrow X$ into a Riemannian manifold

• such an orthonormal local frame $V$ is called adapted or Darboux (generalizing terminology originating in the differential geometry of curves and surfaces, cf. Guggenheimer 1977, p. 210, Petrunin & Barrera 2020, p. 107) for $\phi$ if for each $\sigma \in \Sigma$ and each lift $\widehat{\phi(\sigma)} \in \widehat{X}$ of $\phi(\sigma) \in X$ its first $dim(\Sigma)$ components are tangent to $\Sigma$:

  (1)$$\underset { a \leq dim(\Sigma) } {\forall} \;\;\;\;\; V_a(\sigma) \,\in\, T_\sigma \Sigma \subset T_{\widehat{\phi(\sigma)}} \widehat{X}$$

  (Sternberg 1964, Def. 2.1 on p. 244; Griffiths & Harris 1979 (1.12); Berger, Bryant & Griffiths 1983, p. 818; Yang 1992, p. 157)

• such an orthonormal local co-frame $E$ is called adapted or Darboux (generalizing again the terminology from differential geometry of curves and surfaces, Cartan 1926, p. 211) for $\phi$ if its last $dim(X)-dim(\Sigma)$-components are transversal to $\Sigma$:

  (2)$$\underset { a \gt dim(\Sigma) } {\forall} \;\;\;\;\; \phi^\ast E^a \;=\; 0$$

  (Sternberg 1964, (2.11) on p. 246; Zandi 1988, p. 426; Mastrolia, Rigoli & Setti 2012, Def. 1.17; Giron 2020, §3.2.2; Chen & Giron 2021, §2.4)

Remark

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

$$e \;\coloneqq\; \big( e^a \;\coloneqq\; \phi^\ast E^a \big)_{ a \leq dim(\Sigma) }$$

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

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

(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).

Proposition

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

Proof

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

Hence in this slice coordinate chart a local frame for $X$ is given by the $n$ canonical coordinate vector fields, with the first $k$ of them forming a local frame field on $\Sigma$

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

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

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

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 $\big(v_a^\mu\big)_{a,\mu}$ of components of the above frame (given by $v_a^\mu \partial_\mu = v_a$) is block diagonal (the upper diagonal block being the local frame on $\Sigma$).

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

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

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

This gives the desired coframe field:

$$e^{a} \,\coloneqq\, e^a_\mu \, \mathrm{d}x^\mu$$

which is orthonormal on $X$

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

because

$$v_a^\mu g_{\mu \nu} v_b^\nu \;=\; \delta_{a b} \;\;\;\; \Leftrightarrow \;\;\;\; g_{\mu \nu} \;=\; e^a_\mu \delta_{a b} e^b_\nu \,.$$

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

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

Remark

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

Second fundamental form

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

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

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

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

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

Now let $\Omega$ be the unique torsion-free connection for $E$, in that

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

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

$$\begin{array}{cccl} \omega^a{}_b &\coloneqq& \phi^\ast \Omega^a{}_b & \text{for tangential}\;\text{and tangential}\;b \\ {Ⅱ}^a_{b_1 b_2} e^{b_1} &\coloneqq& \phi^\ast \Omega^a{}_{b_2} & \text{for transversal}\;a\;\text{and tangential}\;b_2 \,. \end{array}$$

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

$$\left\{ \begin{array}{ll} \mathrm{d} e^a - \omega^a{}_b \, e^b \;=\; 0 & \text{for tangential}\;a \\ {Ⅱ}^a_{b_1 b_2} e^{b_1} e^{b_2} \;=\; 0 & \text{for transversal}\;a \end{array} \right.$$

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

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

$${Ⅱ}^a_{b_1 b_2} \;=\; {Ⅱ}^a_{b_2 b_1}$$

is symmetric in its tangential indices $b_i$. This symmetric tensor on $\Sigma$ is called the second fundamental form of the immersion $\phi$.

(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)

References

• Élie Cartan (translated by Vladislav Goldberg from Cartan’s lectures at the Sorbonne in 1926–27): Part E of: Riemannian Geometry in an Orthogonal Frame, World Scientific (2001) [doi:10.1142/4808, pdf]

• Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall (1964), AMS (1983) [ISBNJ:978-0-8218-1385-0, ams:chel-316, ark:/13960/t1pg9dv6k]

• Heinrich W. Guggenheimer, Differential Geometry, Dover (1977) [isbn:9780486634333, ark:/13960/t9t22sk9n]

• Phillip Griffiths, Joseph Harris, Algebraic geometry and local differential geometry, Annales scientifiques de l’École Normale Supérieure, Serie 4, Volume 12 no. 3 (1979) 355-452 [numdam:ASENS_1979_4_12_3_355_0]

• Eric Berger, Robert Bryant, Phillip Griffiths, The Gauss equations and rigidity of isometric embeddings, Duke Math. J. 50 3 (1983) 803-892 [doi:10.1215/S0012-7094-83-05039-1]

• Ahmad Zandi, Minimal immersions of surfaces in quaternionic projective space, Tsukuba Journal of Mathematics 12 2 (1988) 423-440 [jstor:43686661]

• Kichoon Yang, Embeddings of $G$-Structures, chapter VII in: Exterior Differential Systems and Equivalence Problems, Mathematics and its Applications 73, Kluwer (1992), Spinger (1992) [doi:10.1007/978-94-015-8068-7_7]

• Isaac Chavel, Riemannian Submanifolds, §II.2 in: Riemannian geometry – A modern introduction, Cambridge University Press (1993) [doi:10.1017/CBO9780511616822]

• Qing Han, Jia-Xing Hong: Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs 130, AMS (2006) [ISBN:0821840711]

• Paolo Mastrolia, Marco Rigoli, Alberto G. Setti, §1.3 Some formulas for immersed submanifolds in: Yamabe-type Equations on Complete, Noncompact Manifolds, Birkhäuser (2012) [doi:10.1007/978-3-0348-0376-2]

• Tristan Pierre Giron, On the Analysis of Isometric Immersions of Riemannian Manifolds, PhD thesis (2020) [uuid:cae2f41d-c5a1-4138-9dec-5e824d21044e, pdf]

• Gui-Qiang G. Chen, Tristan P. Giron, Weak continuity of curvature for connections in $L^p$ [arXiv:2108.13529]

• Nicholas Kayban, Riemannian Immersions and Submersions (2021) [pdf, pdf]

• Qing Han, Marta Lewicka, Isometric immersions and applications, Notices of the AMS 2023 [arXiv:2310.02566]

• Zuoqin Wang, The method of moving frames, Lecture 11 in: Riemannian geometry (2024) [webpage, pdf, pdf]