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

Key properties of Riemannian immersion — such as whether they are harmonic — are encoded in a tensor called their “second fundamental form”, which is most immediately expressed in terms of a local Darboux coframe field adapted to the immerison.

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.

Local Darboux (co-)frame field

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, we may observe with GSS24, §2 that 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

The “second fundamental form” (on the terminology cf. Rem. below) of a Riemannian immersion $\Sigma \xrightarrow{\phantom{-}} X$ measures the transversal change of tangent vectors to $\Sigma$, under their tangential transport along $\Sigma$ inside $X$.

We first discuss the second fundamental form in terms of local Darboux co-frame fields, where its definition is most immediate, and then extract equivalent expressions in terms of covariant derivatives.

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

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

Then the pullback of the torsion constraint (4) is equivalent to the following 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

(6)$${Ⅱ}^a_{b_1 b_2} \;=\; {Ⅱ}^a_{b_2 b_1}$$

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. Willmore 1996, p. 125-126, Berger, Bryant & Griffiths 1983, p. 819; Chavel 1993 (II.2.12); Wang 2024, Def. 2.2)

Remark

(terminology) Historically, by the “first fundamental form” authors used to refer to the pullback of the metric tensor along an immersion. While this usage is no longer practiced, the term “second fundamental form” for (6) has become standard.

(cf. Chavel 1993, Rem. II..1; Lee 2018, pp. 227)

In much of the literature the second fundamental form is alternatively expressed instead in terms of covariant derivatives of and along tangential vector fields (e.g.: Chavel 1993 §II.2; Baird & Wood 2003, §3.2; Lee 2018, pp. 225), via the following proposition

(which is well-known but seems hard to cite explicitly, cf. maybe Willmore 1996, p. 126):

Proposition

The second fundamental form obtained from a local Darboux coframe field as in (5), when regarded as a tensor, namely as a morphism of vector bundles from the tensor product of the tangent bundle of $\Sigma$ to the normal bundle of $X$ relative to $\phi$

$${Ⅱ} \;\colon\; T\Sigma \otimes T \Sigma \xrightarrow{\;} N_\phi X \,,$$

is given by covariant derivatives of vector fields as follows:

(7)$${Ⅱ}(v,w) \;=\; \nabla^X_{v} \, \mathrm{d}\phi(w) \;-\; \mathrm{d}\phi\big( \nabla^\Sigma_v \, w \big) \,.$$

Proof

Let $(v_a)_{a = 1}^{dim X}$ be a local Darboux frame (1) on $X$, so that $(v_a)_{a =1}^{dim \Sigma}$ is a local frame on $\Sigma$ (called the tangential frame vectors, the others being the transversal ones).

It follows that pushforward of vector field in this Darboux basis is just the canonical injection

(8)$$\begin{array}{rcc} \mathrm{d}\phi_\sigma \;\colon\; T_\sigma \Sigma &\xrightarrow{\phantom{--}}& T_{\phi(\sigma)} X \\ v_a(\sigma) &\mapsto& v_a(\sigma) \end{array}$$

(shown for any $\sigma \in \Sigma$ inside the given chart).

Remembering that for tangential $b_1, b_2$ we set

(9)$$\phi^\ast \Omega_{b_1}{}^a{}_{b_2} \;\equiv\; \left\{ \begin{array}{ll} \omega_{b_1}{}^a{}_{b_2} & \text{tangential}\; a \\ {Ⅱ}^a_{b_1 b_2} & \text{transversal}\; a \end{array} \right.$$

this makes it immediate that

(10)$$\begin{array}{l} \nabla^X_{b_1} \mathrm{d}\phi(v_{b_2}) - \mathrm{d}\phi\big( \nabla^\Sigma_{b_1} v_{b_2} \big) \\ \;=\; \nabla^X_{b_1} v_{b_2} - \nabla^\Sigma_{b_1} v_{b_2} \\ \;=\; \phi^\ast \Omega_{b_1}{}^{a}{}_{b_2} v_a - \omega_{b_1}{}^a{}_{b_2} v_a \\ \;=\; \text{II}^a_{b_1 b_2} v_a \mathrlap{\,.} \end{array}$$

In words, the computation (10) shows that the covariant derivative on $X$ of and along tangent vectors to $\Sigma$ is that on $\Sigma$ plus the contribution of the second fundamental form, hence their difference extracts the latter.

Remark

(further alternative expressions)
As the proof (10) of Prop. shows, the second fundamental form extracts the normal component of the ambient covariant derivative of and along tangential vectors:

$$\text{II}(v,w) \;=\; (\nabla^X_v w)^\perp \,.$$

In this form it appears for instance in Kobayashi & Nomizu 1963 §VII.3, Chavel 1993 (II.2.2); the relation to (7) is made explicit by Baird & Wood 2003 Def. 3.2.3, cf also Willmore 1996, p. 126.

Alternatively, the pushforward of vector fields $\mathrm{d}\phi$ may be understood as a section of the tensor product vector bundle

$$\mathrm{d}\phi \;\in\; \Gamma\big( T\Sigma^\ast \otimes \phi^\ast T X \big) \,,$$

whence the expression (7) may be understood as the induced covariant derivative on that tensor bundle

$$(\nabla \mathrm{d}\phi)(v,w) \;\coloneqq\; \nabla^X_{v} \mathrm{d}\phi(w) \;-\; \mathrm{d}\phi\big( \nabla^\Sigma_v w \big) \,.$$

In this sense the second fundamental form is simply expressed as

$${Ⅱ} \;=\; \nabla \mathrm{d}\phi \,.$$

This is how it appears in Eells & Sampson 1964 p. 123, Baird & Wood 2003 (3.2.1).

Finally, one may write out the covariant derivatives in a coordinate chart in terms of the Christoffel symbols $L_{l}{}^m{}_n$ for the Levi-Civita connection on $X$, and those $\Gamma_i{}^l{}_j$ of the pullback connection to $\Sigma$:

(11)$$\begin{array}{l} {Ⅱ}^k_{i j} \\ \;=\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \nabla^X_{\partial_i} \mathrm{d}\phi( \partial_j ) \;\;\;\;\;\;\;\;\;\;\;\; - \mathrm{d}\phi\big( \nabla^\Sigma_{\partial_i} \partial_j \big) \\ \;=\; \overbrace{ \partial_i \partial_j \phi^l + L_{m}{}^l{}_n (\partial_i \phi^m) (\partial_j \phi^n) } - \overbrace{ \Gamma_{i}{}^k{}_j \partial_k \phi^l } \\ \;=\; \partial_i \partial_j \phi^l - \Gamma_{i}{}^k{}_j \partial_k \phi^l + L_{m}{}^l{}_n (\partial_i \phi^m) (\partial_j \phi^n) \,. \end{array}$$

In this guise the second fundamental form was originally given in Eells & Sampson 1964 (p. 118 & 123 with p. 111) following Eisenhart 1925, §43, reviewed by Baird & Wood 2003 (3.2.2).

Remark

(totally geodesic immersions)
If the fundamental form Ⅱ of a Riemannian immersion $\phi \colon, \Sigma \to X$ vanishes, then geodesics in $\Sigma$ are taken by $\phi$ to geodesics in $X$, hence such $\phi$ is said to be totally geodesic [Eells & Sampson 1964 p. 126, Baird & Wood 2003 Def. 3.2.1].

Tension field

Definition

The tension field of the immersion $\phi$ is the contraction (trace) of the second fundamental form:

$$\tau^k \;\coloneqq\; \eta^{a b} \, {Ⅱ}^k_{a b} \;=\; g^{i j} \, {Ⅱ}^k_{i j} \,.$$

From (11) we have immediately the coordinate expression

$$\begin{array}{l} \tau^k \;=\; \eta^{a b} \, {Ⅱ}^k_{a b} \;=\; g^{i j} \, {Ⅱ}^k_{i j} \\ \;=\; \underset {\Delta \phi^l} { \underbrace{ g^{i j} \big( \partial_i \partial_j \phi^l - \Gamma_{i}{}^k{}_j \partial_k \phi^l \big) } } + g^{i j} (\partial_i \phi^m) (\partial_j \phi^n) L_{m}{}^l{}_n \,, \end{array}$$

where “$\Delta$” denotes the Laplace operator on $\Sigma$.

(Eells & Sampson 1964 (5), Baird & Wood 2003 Def. 3.2.4)

Harmonic immersions

Theorem

(harmonic equation)
The vanishing of the tension field $\tau$ (Def. ) characterizes Riemannian immersion which are harmonic maps.

(Eells & Sampson 1964 pp. 116, Baird & Wood 2003 Thm. 3.3.3)

Literature

• Luther P. Eisenhart, Chapter IV of: Riemannian Geometry, Princeton University Press (1925, 1950) [ISBN:9780691023533, ark:/13960/t47q47r0k, pdf]

• 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]

• Shoshichi Kobayashi, Katsumi Nomizu, Chapter VII in: Foundations of differential geometry, Volume 1 (1963), Volume 2 (1969) Interscience Publishers, reprint: Wiley Classics Library (1996) [ISBN:978-0-470-55558-3, Wikipedia entry]

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

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

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

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• 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]