Flat approximations of hypersurfaces along curves

TL;DR

This paper investigates the existence and uniqueness of flat surface approximations along curves in higher-dimensional surfaces, providing explicit constructions based on classical characterizations of flat surfaces.

Contribution

It introduces a method to construct flat surface approximations along curves in higher dimensions using explicit parametric formulas.

Findings

Existence and uniqueness results for flat approximations along curves.

Explicit parametric construction based on torses.

Extension of classical flat surface characterizations to higher dimensions.

Abstract

Given a smooth curve $\gamma$ in some $m$-dimensional surface $M$ in $\mathbb{R}^{m+1}$, we study existence and uniqueness of a flat surface $H$ having the same field of normal vectors as $M$ along $\gamma$, which we call a flat approximation of $M$ along $\gamma$. In particular, the well-known characterisation of flat surfaces as torses (ruled surfaces with tangent plane stable along the rulings) allows us to give an explicit parametric construction of such approximation.

Equations

$$⟨X(v_1,\dots ,v_k),v_i⟩=0,1\le i\le k.$$

$$⟨X(v_1,\dots ,v_k),X(v_1,\dots ,v_k)⟩=\det (⟨v_i,v_j⟩).$$

$$\left(v_1,\dots ,v_{n-1},X(v_1,\dots ,v_{n-1})\right)\in \mathcal{O}.$$

$$v_1\times \cdots \times v_{n-1}=\star (v_1\wedge \cdots \wedge v_{n-1})$$

$$L^{k}TM=\underset{p\in M}{⨆}{L}^k(T_{p}M).$$

$$X_1\times \cdots \times X_{m-1}=\star (X_1\wedge \cdots \wedge X_{m-1}).$$

$${\overline{\nabla }}_X{\rm Y}=({\overline{\nabla }}_X{\rm Y}{)}^{\top }+({\overline{\nabla }}_X{\rm Y}{)}^{\perp },$$

$$(X,Y)\mapsto ({\overline{\nabla }}_{X}Y{)}^{\top }$$

$$(X,N)\mapsto ({\overline{\nabla }}_{X}N{)}^{\perp }.$$

$$W_{m+1}=E_1\times \cdots \times E_m,$$

$${\overline{D}}_t{E}_i=D_t{E}_i+{\tau }_i{W}_{m+1},$$

$${\overline{D}}_t{W}_{m+1}=-{\tau }_1{E}_1-\cdots -{\tau }_m{E}_m.$$

$$\sigma (t,u)=\gamma (t)+uX(t)$$

$$\sigma (t,u)=\gamma (t)+{\sum }_j{u}^j{X}_j(t)$$

$$Z=\frac{\partial \sigma }{\partial t}\times \frac{\partial \sigma }{\partial u^1}\times \cdots \times \frac{\partial \sigma }{\partial u^{m-1}},$$

$${\overline{\nabla }}_{X}N=0.$$

$${\partial }_1\hat{N}=\cdots ={\partial }_{m-1}\hat{N}=0,$$

$$\begin{array}{rl}\dot{\gamma }\cdot {\partial }_{1}Z\equiv \dot{\gamma }\cdot {\overline{D}}_t{X}_1\times X_1\times \cdots \times X_{m-1}& =0\\ & \text{makebox}[0.0pt]⋮\\ \dot{\gamma }\cdot {\partial }_{m-1}Z\equiv \dot{\gamma }\cdot {\overline{D}}_t{X}_{m-1}\times X_1\times \cdots \times X_{m-1}& =0\end{array}$$

$$Z(t,u)=\{\dot{\gamma }(t)+u^i{\overline{D}}_t{X}_i(t)\}\times X_1(t)\times \cdots \times X_{m-1}(t),$$

$$|Z|({\partial }_{j}Z{)}^{\top }-Z^{\top }{\partial }_j|Z|=0,$$

$${\partial }_j\hat{N}=\frac{({\partial }_{j}Z{)}^{\top }|Z|-Z^{\top }{\partial }_j|Z|}{|Z{|}^2}.$$

$$\{(X_1,\dots ,X_n)\sim (Y_1,\dots ,Y_n)\}\iff \{\mathrm{span}(X_1,\dots ,X_n)=\mathrm{span}(Y_1,\dots ,Y_n)\}.$$

$$\dot{\gamma }\cdot {\overline{D}}_t{X}_j\times X_1\times \cdots \times X_{m-1}=0$$

$$\dot{\gamma }(t)\times X_1(t)\times \cdots \times X_{m-1}(t)\ne 0$$

$$E_1\cdot ({\overline{D}}_t{X}_j^i{E}_i+X_j^i{\overline{D}}_t{E}_i)\times X_1^i{E}_i\times \cdots \times X_{m-1}^i{E}_i=0,$$

$$\sum _{i=1}^m{\overline{D}}_t{E}_i$$

$$=\sum _{i=1}^m\left\{(D_t{E}_i\cdot E_1)E_1+\cdots +(D_t{E}_i\cdot E_m)E_m\right\}+E_{m+1}\sum _{i=1}^m{\tau }_i.$$

$$E_1\cdot X_j^i{\tau }_i{E}_{m+1}\times (X_1^2{E}_2+\cdots +X_1^m{E}_m)\times \cdots \times (X_{m-1}^2{E}_2+\cdots +X_{m-1}^m{E}_m)=0.$$

$$-X_j^i{\tau }_i\sum _{\sigma \in S_m^2}\mathrm{sgn}(\sigma )X_1^{\sigma (2)}\cdots X_{m-1}^{\sigma (m)}=0.$$

$$X_j(Z)=Z\times E_2\times \cdots \times {\tilde{E}}_{m-j+1}\times \cdots \times E_m,$$

$$X_j^i(Z){\tau }_i=(-1{)}^j{Z}^{m-j+1}{\tau }_1+(-1{)}^{j-1}{Z}^1{\tau }_{m-j+1}.$$

$$\begin{array}{rl}{Z}^m{\tau }_1-Z^1{\tau }_m& =0\\ Z^{m-1}{\tau }_1-Z^1{\tau }_{m-1}& =0\\ & \text{makebox}[0.0pt]⋮\\ Z^3{\tau }_1-Z^1{\tau }_3& =0\\ Z^2{\tau }_1-Z^1{\tau }_2& =0.\end{array}$$

Full text

1. Introduction and Main Result

Developable, or flat, hypersurfaces in $\mathbb{R}^{m+1}$, where $m\geq 2$, are classical objects in Riemannian geometry. They are characterised by being foliated by open subsets of $(m-1)$-dimensional planes, called rulings, along which the tangent space remains stable [14, Theorem 1]. Here we are concerned with the problem of existence and uniqueness—as well as with the explicit construction—of flat approximations of hypersurfaces along curves. Let $M^{m}$ be a (possibly curved) Euclidean hypersurface and $\gamma$ a curve in $M^{m}$. A hypersurface $H$ is called an approximation of $M^{m}$ along $\gamma$ if the two manifolds have common tangent space at every point of $\gamma$.

In dimension $2$, the question of existence has been settled for a long time. A constructive proof, under suitable assumptions, is already present in Do Carmo’s textbook [5, p. 195–197]. It turns out the existence of a flat approximation of $M^{2}$ along $\gamma$ implies the existence of a rolling, in Nomizu’s sense, of $M^{2}$ on the tangent space $T_{\gamma(0)}M^{2}$ along the given curve – see [9] and [11]. More recently, Izumiya and Otani have shown uniqueness [6, Corollary 6.2].

In this paper, we extend the result in [5] to any curve in $M^{m}$. More precisely, we shall present a constructive proof of the following

Theorem 1.1**.**

Let $\gamma\colon I\to M^{m}$ be a smooth curve in a hypersurface $M^{m}$ in $\mathbb{R}^{m+1}$. If the curve is never parallel to an asymptotic direction of $M^{m}$, then there exists a flat approximation $H$ of $M^{m}$ along $\gamma$. Such hypersurface is unique in the following sense: if $H_{1}$ and $H_{2}$ are two flat approximations of $M^{m}$ along $\gamma$, then they agree on an open set containing $\gamma(I)$.

The strategy to prove this result involves looking for $(m-1)$-tuples of linearly independent vector fields $(X_{1},\dotsc,X_{m-1})$ along $\gamma$ satisfying $\dot{\gamma}(t)\notin\operatorname{span}(X_{j}(t))_{j\,=\,1}^{m-1}$ for all $t$ and having zero normal derivative (normal projection of Euclidean covariant derivative). Indeed, such conditions guarantee the image of the map $\gamma+\operatorname{span}(X_{j})_{j\,=\,1}^{m-1}$ be a flat hypersurface of $\mathbb{R}^{m+1}$ in a neighbourhood of $\gamma$. The main difficulty resides in getting around the many-to-one correspondence between tuples of vector fields and rank-$(m-1)$ distributions along $\gamma$.

It is worth pointing out that the solution depends on the original hypersurface $M^{m}$ only through its distribution of tangent planes along $\gamma$. Thus, when $m=2$, our problem is nothing but the classical Björling’s problem—to find all minimal surfaces passing through a given curve with prescribed tangent planes—addressed to a different class of surfaces. In this respect, the present work joins several other recent studies aimed at solving Björling-type questions, see [3, 2] and references therein.

The paper is organised as follows. The next two sections present some preliminaries, mostly for the sake of introducing relevant notation and terminology. In Section $4$ we derive a simple condition for discerning when a parametrised ruled hypersurface has a flat metric. Such condition is then used in Section $5$ to prove the main theorem. Finally, in Section $6$ we give some general remarks about the construction of the approximation.

2. Vector Cross Products

Let $V$ be an $n$-dimensional, real vector space equipped with a positive definite inner product $\langle\cdot{,}\cdot\rangle$. In the following, $V^{k}$ will indicate the $k$-th Cartesian power of $V$, and $L^{k}(V)$ the set of all multilinear maps from $V^{k}$ to $V$. Note that, under pointwise addition and scalar multiplication, $L^{k}(V)$ is a finite dimensional vector space, in that it is naturally isomorphic to the space $T^{(1,k)}(V)$ of tensors on $V$ of type $(1,k)$ – see for example [8, Lemma 2.1]. Thus, $\dim L^{k}(V)=n^{k+1}$.

A $k$-fold vector cross product on $V$, $1\leq k\leq n$, is an element of $L^{k}(V)$—i.e., a multilinear map $X\colon V^{k}\to V$—satisfying the following two axioms:

[TABLE]

We emphasize that the second axiom implies any such $X$ being alternating.

In particular, in the case $V$ carries an orientation $\mathcal{O}$, we say that an $(n-1)$-fold vector cross product $X$ is positively oriented if the following condition holds for all $(n-1)$-tuples of linearly independent vectors $v_{1},\dotsc,v_{n-1}$:

[TABLE]

Analogously, a negatively oriented vector cross product satisfies the same relation with $-\mathcal{O}$ in place of $\mathcal{O}$.

In [4], Brown and Gray proved the following theorem:

Theorem 2.1**.**

Let $V$ be an oriented finite dimensional inner product space, of dimension $n$. There exists a unique positively oriented $(n-1)$-fold vector cross product $X=\cdot\times\dotsb\times\cdot$ on $V$. It is given by:

[TABLE]

where $\star$ is the Hodge star operator on $V$.

We now turn our attention to manifolds. If $M$ is a smooth Riemannian manifold of dimension $m$, let $L^{k}\mathit{TM}$ be the disjoint union of all the vector spaces $L^{k}(T_{p}M)$:

[TABLE]

Clearly, for $L^{k}(T_{p}M)\cong T^{(1,k)}(T_{p}M)$, the set $L^{k}\mathit{TM}$ has a canonical choice of topology and smooth structure turning it into a smooth vector bundle of rank $m^{k+1}$ over $M$. We define a $k$-fold vector cross product on $M$, where $1\leq k\leq m$, to be a smooth section $X$ of $L^{k}\mathit{TM}$ such that, for every point $p\in M$, the map $X_{p}$ is a $k$-fold vector cross product on $T_{p}M$.

We thus have the following corollary of Theorem 2.1:

Corollary 2.2**.**

Let $M$ be a smooth oriented $m$-dimensional Riemannian manifold. There exists a unique $(m-1)$-fold positively oriented vector cross product on $M$. It acts on $(m-1)$-tuples of vector fields $X_{1},\dotsc,X_{m-1}$ on $M$ by

[TABLE]

3. Frames Along Curves

In this section we review some basic facts about Euclidean submanifolds and orthonormal frames along curves.

Let us start with some notation. If $m\geq 2$, let $M$ be an $m$-dimensional embedded submanifold of $\mathbb{R}^{d}$, and $\gamma\colon I=[0,\alpha]\to M$ a smooth regular curve in $M$. Throughout this paper, $\mathbb{R}^{d}$ will always be equipped with the standard Euclidean metric $\overline{g}$, typically indicated by a dot “$\,\cdot\,$”, and standard orientation. Thus, there is a natural choice of Riemannian metric on $M$: the induced metric $\iota^{\ast}\overline{g}$, i.e., the pullback of $\overline{g}$ by the inclusion $\iota\colon M\hookrightarrow\mathbb{R}^{d}$.

Working with submanifolds, it is customary to identify each tangent space $T_{p}M$ with its image under the differential of $\iota$. In so doing, the ambient tangent space $T_{p}\mathbb{R}^{d}$ splits as the orthogonal direct sum $T_{p}M\oplus N_{p}M$, where $N_{p}M$ is the normal space of $M$ at $p$. Thus, the set $\mathfrak{X}(M)$ of tangent vector fields on $M$ becomes a proper subset of the set of vector fields along $M$, which we denote by $\overline{\mathfrak{X}}(M)$. If $X\in\mathfrak{X}(M)$ and $\varUpsilon \in\overline{\mathfrak{X}}(M)$,

[TABLE]

where $\overline{\nabla}$ is the Euclidean connection, $\top$ and $\perp$ are the orthogonal projections onto the tangent and normal bundle of $M$, and where the vector fields $X$ and $\varUpsilon$ are extended arbitrarily to $\mathbb{R}^{d}$. It turns out that the map $\mathfrak{X}(M)\times\mathfrak{X}(M)\to\mathfrak{X}(M)$ defined by

[TABLE]

is a linear connection on $M$, called the tangential connection. In fact, it is no other than the (intrinsic) Levi-Civita connection $\nabla$ of $(M,\iota^{\ast}\overline{g})$.

Similarly, indicating by $\mathfrak{X}(M)^{\perp}$ the set of normal vector fields along $M$, we define the normal connection on $M$ as the map $\mathfrak{X}(M)\times\mathfrak{X}(M)^{\perp}\to\mathfrak{X}(M)^{\perp}$ given by

[TABLE]

Let us recall that an orthonormal frame along $\gamma$ is an $m$-tuple of smooth vector fields $(E_{i})_{i\,=\,1}^{m}$ along $\gamma$ such that $(E_{i}(t))_{i\,=\,1}^{m}$ is an orthonormal basis of $T_{\gamma(t)}M$ for all $t$. In particular, an orthonormal frame $(W_{1},\dotsc,W_{d})$ along a curve $\iota\circ\gamma$ in $\mathbb{R}^{d}$ is said to be $M$-adapted if $(W_{i})_{i\,=\,1}^{m}$ spans the ambient tangent bundle over $\gamma$.

In the remainder of this section, we assume that $M$ has codimension one in $\mathbb{R}^{d}$, i.e., that $d=m+1$. Under such hypothesis, given any orthonormal frame $(E_{i})_{i\,=\,1}^{m}$ along $\gamma$, we can construct an associated $M$-adapted orthonormal frame along $\iota\circ\gamma$ as follows. For $k=1,\dotsc,m$, let $W_{k}=E_{k}$; then, for $k=m+1$,

[TABLE]

so that $(W_{1},\dotsc,W_{m+1})$ is the unique extension of $(E_{i}(t))_{i\,=\,1}^{m}$ to a positively oriented, orthonormal frame along $\iota\circ\gamma$.

Denoting by $D_{t}$ and $\overline{D}_{t}$ the covariant derivative operators determined by $\nabla$ and $\overline{\nabla}$, respectively, we may write

[TABLE]

for some smooth function $\tau_{i}\colon I\to\mathbb{R}$. Clearly, should $M$ be orientable, $\tau_{i}=\pm h(E_{1},E_{i})$, where $h$ is the (scalar) second fundamental form of $M$ determined by a choice of unit normal vector field. Moreover, it easily follows from orthonormality that

[TABLE]

4. Developable Surfaces

The main purpose of this section is to generalize to higher dimensions the following well-known fact about ruled surfaces in $\mathbb{R}^{3}$ – see for example [5, p. 194]:

Lemma 4.1**.**

Let $I$, $J$ be open intervals. Further, let $\gamma$ and $X$ be curves $I\to\mathbb{R}^{3}$ such that the map $\sigma\colon I\times J\to\mathbb{R}^{3}$ given by

[TABLE]

is a smooth injective immersion. Then the Gauss curvature of $\sigma(I\times J)$ is zero precisely when $\gamma$ and $X$ satisfy $\dot{\gamma}\cdot\dot{X}\times X=0$.

We shall begin with some definitions extending the classical notions of ruled and torse surface to arbitrary dimension, yet keeping the codimension fixed to $1$. If $m\geq 2$, let $H$ be a hypersurface in $\mathbb{R}^{m+1}$, as always smooth and embedded.

Definition 4.2**.**

We say that $H$ is a ruled surface if

1. (1)

$H$ is free of planar points, that is, there exists no point of $H$ where the second fundamental form vanishes; 2. (2)

there exists a ruled structure on $H$, that is, a foliation of $H$ by open subsets of $(m-1)$-dimensional affine subspaces of $\mathbb{R}^{m+1}$, called rulings.

In particular, a ruled surface $H$ is said to be a torse surface if, for every pair of points $(p,q)$ on the same ruling, we have $T_{p}H=T_{q}H$, i.e., if all tangent spaces of $H$ along a fixed ruling can be canonically identified with the same linear subspace of $\mathbb{R}^{m+1}$.

Remark 4.3*.*

Although condition 1 in Definition 4.2 may seem overly restrictive, it gives any ruled surface $H$ a desirable property. Namely, it ensures the existence of a smooth ruled parametrisation of $H$ [13]. On the other hand, we will also need to work with the broader class of generalised ruled hypersurfaces obtained by relaxing such condition. It is well known that every generalised torse with planar points is made up of both standard torses and pieces of $m$-planes, always glued along a well-defined ruling.

Remember that any $d$-dimensional Riemannian manifold locally isometric to $\mathbb{R}^{d}$ is said to be flat. In particular, the classical term for hypersurfaces is developable, see [14, Section 1] for a detailed discussion on terminology. Remarkably, it turns out that

Theorem 4.4** ([14, Theorem 1]).**

$H$* is a torse surface if and only if it is free of planar points and, when equipped with the induced metric $\iota^{\ast}\overline{g}$, $H$ becomes a flat Riemannian manifold.*

Corollary 4.5**.**

$H$* is a generalised torse surface if and only if the induced metric on $H$ is flat.*

Given a curve $\gamma$ in $\mathbb{R}^{m+1}$, the following result is key for constructing ruled surfaces containing $\gamma$. Note that in its statement we use the canonical isomorphism between $\mathbb{R}^{m+1}$ and any of its tangent spaces to identify the vector fields $X_{1},\dotsc,X_{m-1}$ along $\gamma$ with curves in $\mathbb{R}^{m+1}$.

Lemma 4.6**.**

Let $I$ be a closed interval. Let $\gamma\colon I\to\mathbb{R}^{m+1}$ be a smooth injective immersion. Let $(X_{1},\dotsc,X_{m-1})$ be a smooth, linearly independent $(m-1)$-tuple of vector fields along $\gamma$ such that $\dot{\gamma}(t)\times X_{1}(t)\times\dotsb\times X_{m-1}(t)\neq 0$ for all $t\in I$. Then there exists an open box $V$ in $\mathbb{R}^{m-1}$ containing the origin such that the restriction to $I\times V$ of the map $\sigma\,\colon I\times\mathbb{R}^{m-1}\to\mathbb{R}^{m+1}$ defined by

[TABLE]

is a smooth embedding.

Proof.

To show that $\sigma$ restricts to an embedding, we first prove the existence of an open box $V_{1}$ such that $\sigma\rvert_{I\times V_{1}}$ is a smooth immersion. Essentially, the statement will then follow by compactness of $I$.

Obviously, $\sigma$ is immersive at $(t,u)$ if and only if the length $\ell\,\colon I\times\mathbb{R}^{m-1}\to\mathbb{R}$ of the cross product of the partial derivatives of $\sigma$ is non-zero at $(t,u)$. Thus, define $W_{t}$ to be the subset of $\{t\}\times\mathbb{R}^{m-1}$ where $\sigma$ is immersive. It is an open subset in $\mathbb{R}^{m-1}$ because it is the inverse image of an open set under a continuous map, $W_{t}=\ell(t,\cdot)^{-1}(\mathbb{R}\setminus\{0\})$; it contains [math] by assumption. Thence, there exists an $\epsilon_{t}>0$ such that the open ball $B(\epsilon_{t},0)\subset\mathbb{R}^{m-1}$ is completely contained in $W_{t}$. Letting $\epsilon_{1}=\inf_{t\,\in\,I}(\epsilon_{t})$, we can conclude that the restriction of $\sigma$ to the box $I\times(-\epsilon_{1}/2,\epsilon_{1}/2)^{m-1}$ is a smooth immersion.

Now, being $\sigma$ a smooth immersion on $I\times V_{1}$, it follows that every point of $I\times V_{1}$ has a neighbourhood on which $\sigma$ is a smooth embedding. Let then $W_{t}^{\prime}$ be the subset of $W_{t}$ where $\sigma$ is an embedding. It is open in $\mathbb{R}^{m-1}$, and it contains the origin because $\gamma$ is a smooth injective immersion of a compact manifold. From here we may proceed as before. ∎

Thus, for suitably chosen $(X_{1},\dotsc,X_{m-1})$ and $V\subset\mathbb{R}^{m-1}$, we have verified that $H_{\sigma}=\operatorname{Im}\sigma\rvert_{I\times V}$ is a hypersurface in $\mathbb{R}^{m+1}$, and $\mathscr{F}_{\sigma}=\{\sigma(t,V)\}_{t\,\in\,I}$ a ruled structure on it. Under such hypothesis, let us assume $H_{\sigma}$ is orientable (this we can do, possibly limiting the analysis to an open subset). Then, we may pick out a smooth unit normal vector field $N$ along $H_{\sigma}$ by means of the $m$-fold cross product on $\mathbb{R}^{m+1}$, as follows. Letting

[TABLE]

define $\widehat{N}=Z\lvert Z\rvert^{-1}$, and so $N=\widehat{N}\circ\sigma^{-1}$. In this situation, assuming there are no planar points, $H_{\sigma}$ being a torse surface is equivalent to $N$ being constant along each of the rulings. Thus, indicating with $\overline{\nabla}$ the Euclidean connection on $\mathbb{R}^{m+1}$, $(H_{\sigma},\iota^{\ast}\overline{g})$ is flat if and only if, for all vector fields $X$ tangent to $\mathscr{F}_{\sigma}$ on $H_{\sigma}$:

[TABLE]

In fact, by linearity – and writing $\partial_{j}$ as a shorthand for $\frac{\partial}{\partial u^{j}}$ – it suffices that (3) holds for the vector fields $\sigma_{\ast}(\partial_{1}),\dotsc,\sigma_{\ast}(\partial_{m-1})$ spanning the distribution corresponding to $\mathscr{F}_{\sigma}$. We may thereby express the developability condition for $(H_{\sigma},\iota^{\ast}\overline{g})$ simply as

[TABLE]

where we understand $\partial_{j}$ as acting on the coordinate functions $\widehat{N}^{1},\dotsc,\widehat{N}^{m+1}$ of $\widehat{N}$ in the standard coordinate frame of $T\mathbb{R}^{m+1}$.

The next lemma finally translates (4) into $m-1$ conditions involving the vector fields $X_{1},\dotsc,X_{m-1}$ along $\gamma$, and represents the sought generalization of Lemma 4.1. It says that $\iota^{\ast}\overline{g}$ is a flat Riemannian metric precisely when $\overline{D}_{t}X_{j}=D_{t}X_{j}$ for every $j$, or equivalently when each of the normal projections $(\overline{D}_{t}X_{1})^{\perp},\dotsc,(\overline{D}_{t}X_{m-1})^{\perp}$ vanishes identically.

Lemma 4.7**.**

Assume $\sigma\rvert_{I\times V}$ is a smooth embedding. The hypersurface $H_{\sigma}$ is a generalised torse surface if and only if the following equations hold:

[TABLE]

Proof.

Computing the partial derivatives of $\sigma$ and substituting them into the expression (2) for $Z$, we get:

[TABLE]

from which the identity $\partial_{j}Z\equiv\overline{D}_{t}X_{j}\times X_{1}\times\dotsb\times X_{m-1}$ clearly follows. Thus, we need to prove that $\partial_{1}\widehat{N}=\dotsb=\partial_{m-1}\widehat{N}=0$ if and only if $\partial_{1}Z\cdot\dot{\gamma}=\dotsb=\partial_{m-1}Z\cdot\dot{\gamma}=0$. In fact, for $\partial_{j}Z$ is orthogonal to $X_{1},\dotsc,X_{m-1}$, it is enough to check that $\partial_{1}\widehat{N}=\dotsb=\partial_{m-1}\widehat{N}=0$ if and only if $(\partial_{1}Z)^{\top}=\dotsb=(\partial_{m-1}Z)^{\top}=0$. First, assume $\partial_{j}\widehat{N}=0$. Since $\widehat{N}=Z\lvert Z\rvert^{-1}$, it follows by linearity of the tangential projection that

[TABLE]

which is true exactly when $(\partial_{j}Z)^{\top}=0$, as desired. To verify the converse, note that $(\partial_{j}N)^{\perp}=0$ because $N$ has unit length. Thus, again by linearity of $\top$,

[TABLE]

Since $Z^{\top}=0$, the claim follows. ∎

5. Proof of the Main Result

Here we prove our main result, stated in Theorem 1.1 in the Introduction. The proof is constructive and is based on the fact that an Euclidean hypersurface without planar points has a flat induced metric precisely when it is a torse surface (Theorem 1.3). Let $M$ be a hypersurface in $\mathbb{R}^{m+1}$ and $\gamma$ a smooth curve in $M$, as defined at the beginning of Section $3$. Denoting by $\mathfrak{X}(\gamma)$ the set of smooth, non-vanishing vector fields along $\gamma$, define an equivalence relation on the $n$-th Cartesian power $\mathfrak{X}(\gamma)^{n}$ of $\mathfrak{X}(\gamma)$ by the following rule:

[TABLE]

Let us indicate an element of the quotient $\mathfrak{X}(\gamma)^{n}/{\sim}$, that is, an element of $\mathfrak{X}(\gamma)^{n}$ up to equivalence, by $[X_{1},\dotsc,X_{n}]$. We wish to find $[X_{1},\dotsc,X_{m-1}]$ such that, for every $t\in I$ and integer $j$ with $1\leq j\leq m-1$, both the conditions

[TABLE]

are satisfied. Beware that, throughout this section, we will extensively use Einstein summation convention: every time the same index appears twice in any monomial expression, once as an upper index and once as a lower index, summation over all possible values of that index is understood.

Once and for all, let us choose a $\gamma$-adapted orthonormal frame $(E_{1},\dotsc,E_{m})$ along $\gamma$: this is just an orthonormal frame along $\gamma$ whose first element coincides with the tangent vector $\dot{\gamma}$. The first step is to rewrite (6) as an equation involving the $m(m-1)$ coordinate functions $X_{j}^{i}$ of $X_{1},\dotsc,X_{m-1}$ with respect to $(E_{1},\dotsc,E_{m})$. Differentiating covariantly $X_{j}=X_{j}^{i}E_{i}$ and substituting, we obtain

[TABLE]

whereas, from (1).

[TABLE]

Now, given any ordered $m$-tuple $(i_{1},\dotsc,i_{m})$ of integers with $1\leq i_{1}\leq m+1$ and $1\leq i_{k}\leq m$ for $k=2,\dotsc,m$, a necessary condition for the $m$-fold cross product $E_{i_{1}}\times\dotsb\times E_{i_{m}}$ to give either $E_{1}$ or $-E_{1}$ is that $i_{1}=m+1$ and $i_{k}\neq 1$. It follows that (8) is equivalent to

[TABLE]

In fact, $E_{i_{1}}\times\dotsb\times E_{i_{m}}=\pm E_{1}$ if and only if $i_{1}=m+1$ and the $(m-1)$-tuple $(i_{2},\dotsc,i_{m})$ is a permutation of $(2,\dotsc,m)$. In particular, if it is an even permutation, then the basis $(E_{m+1},E_{i_{2}},\dotsc,E_{i_{m}},E_{1})$ is negatively oriented, for transposing $E_{m+1}$ and $E_{1}$ must give a positive basis, and so $E_{i_{1}}\times\dotsb\times E_{i_{m}}=-E_{1}$. Thence, denoting by $S_{m}^{2}$ the group of permutations $\sigma$ of $(2,\dotsc,m)$, we may write (9) simply as

[TABLE]

On the other hand, a similar computation would reveal that condition (7) is satisfied for every $t$ if and only if the summation term above (the term independent of $j$) never vanishes. We may thereby conclude that, under the assumption of (7) being true, condition (6) is equivalent to $X_{j}^{i}\tau_{i}=0$.

Next, consider the set $\mathscr{Z}\subset\mathfrak{X}(\gamma)$ of smooth vector fields $Z$ along $\gamma$ such that $Z^{1}(t)=Z\cdot E_{1}(t)\neq 0$ for every $t$. We establish a bijection between its quotient $\mathscr{Z}/{\sim}$ by $\sim$ and the subset of $\mathfrak{X}(\gamma)^{m-1}/{\sim}$ where (7) holds. For every $j$, let

[TABLE]

where the tilde indicates that $E_{m-j+1}$ is omitted, so that the cross product is $(m-1)$-fold. For example, when $j=1$, we omit the last vector field $E_{m}$; when $j=2$ the second to last, and so on, until dropping $E_{2}$ for $j=m-1$. Linear independence of $E_{1},X_{1}(Z),\dotsc,X_{m-1}(Z)$ is easily seen, as by definition $Z$ is never in the span of $E_{2},\dotsc,E_{m}$. Since the normal projection $Z\mapsto Z^{\perp}$ induces a bijection between $\mathscr{Z}/{\sim}$ and the set of smooth $(m-1)$-distributions along $\gamma$ nowhere parallel to $E_{1}$, it follows that the map $[Z]\mapsto[X_{1}(Z),\dotsc,X_{m-1}(Z)]$ between classes of equivalence is indeed a valid parametrisation of the solution set of (7).

We then compute the coordinates of the cross product in (10) with respect to the frame $(E_{1},\dotsc,E_{m})$. Substituting $Z=Z^{i}E_{i}$, all but the terms $Z^{1}E_{1}$ and $Z^{m-j+1}E_{m-j+1}$ will not give any contribution. In particular, $E_{1}\times\dotsb\times\widetilde{E}_{m-j+1}\times\dotsb\times E_{m}=\pm E_{m-j+1}$ depending on whether $(E_{1},\dotsc,\widetilde{E}_{m-j+1},\dotsc,E_{m},E_{m-j+1})$ is positively or negatively oriented. Since the corresponding permutation of $(1,\dotsc,m)$ has sign $(-1)^{j-1}$, we conclude that $X_{j}^{m-j+1}(Z)=(-1)^{j-1}Z^{1}$. An analogous argument would show that $X_{j}^{1}(Z)=(-1)^{j}Z^{m-j+1}$.

Summing up, solving the original problem on $\mathfrak{X}(\gamma)^{m-1}/{\sim}$ essentially amounts to finding $[Z]\in\mathscr{Z}/{\sim}$ such that $X_{j}^{i}(Z)\tau_{i}=0$ for every $j$. Moreover, by the previous computation,

[TABLE]

Thus, denoting again by $\sim$ the equivalence relation on $C^{\infty}(I)^{m}=C^{\infty}(I;\mathbb{R}^{m})$ naturally induced from the one on $\mathfrak{X}(\gamma)$, we need to look for $(Z^{1},\dotsc,Z^{m})$, up to equivalence, satisfying the following system of $m-1$ linear equations on $C^{\infty}(I;\mathbb{R}_{\neq 0})\times C^{\infty}(I)^{m-1}$:

[TABLE]

Assume $\tau_{1}(t)\neq 0$ for all $t$. Then, for any given $Z^{1}$ (remember $Z^{1}$ is non-vanishing by definition), the system has solution

[TABLE]

However, it is easy to see that all solutions are in one and the same equivalence class. Indeed, if $f$ and $g$ are two distinct values of $Z^{1}$, then

[TABLE]

In particular, letting $Z^{1}=\tau_{1}$, we obtain $Z^{i}=\tau_{i}$ for every $i=1,\dotsc,m$, and the solution of the original problem on $\mathfrak{X}(\gamma)^{m-1}/{\sim}$ is given by

[TABLE]

As for uniqueness, in view of Remark 4.3, it is sufficient to show that the condition $\tau_{1}(t) \neq 0$ for all $t$ implies any flat approximation $H$ of $M^{m}$ along $\gamma$ be free of planar points, i.e., be a torse surface. To see this, let $N_{H}$ and $N_{M}$ be smooth unit normal vector fields along $H$ and $M^{m}$, respectively, defined in a neighbourhood of $\gamma(t)$. Then, $\overline{D}_{t}N_{H}=\overline{D}_{t}N_{M}$. Since $H$ is a generalised torse surface by Corollary 4.5, the claim easily follows.

6. Construction of an Adapted Frame

As seen in the last section, the construction of the flat approximation of $M$ along $\gamma$ requires choosing some $\gamma$-adapted orthonormal frame $(E_{i})_{i\,=\,1}^{m}$ along $\gamma$. We emphasize that such a choice is completely arbitrary. If the curve in question satisfies some (rather strong) conditions on its derivatives, then a natural generalization of the classical Frenet–Serret frame is available. The reader may find details on this construction in [12] or [7]. Here we briefly review an alternative approach, one that does not require any initial assumption on the curve. Such approach is due to Bishop [1].

First of all, since the problem is local, we are free to assume that $\gamma$ is a smooth embedding. Thus, for any point $p\in S=\gamma(I)$, there exist slice coordinates $(x_{1},\dotsc,x_{m})$ in a neighbourhood $U$ of $p$. It follows that $(\partial_{1}\lvert_{p},\dotsc,\partial_{m}\lvert_{p})$ is a $\gamma$-adapted basis of $T_{p}M$, i.e., it satisfies $T_{p}S=\operatorname{span}\partial_{1}\lvert_{p}$ and $N_{p}S= \operatorname{span}(\partial_{2}\lvert_{p},\dotsc,\partial_{m}\lvert_{p})$. By applying the Gram–Schmidt process to these vectors, one obtains an orthonormal basis $(n_{j})$ of $N_{p}S$. Although this basis is by no means canonical, the normal connection $\nabla^{\perp}$ of $S$ provides an obvious means for extending it to a frame for the normal bundle of $S$: for each $j$, let $\varUpsilon_{j}$ be the unique normal parallel vector field along $\gamma$ such that $\varUpsilon_{j}\lvert_{p}=n_{j}$ – see [10, p. 119]. Because normal parallel translation is an isometry, the frame $(\dot{\gamma},\varUpsilon_{1},\dotsc,\varUpsilon_{m-1})$ is an orthonormal adapted frame along $\gamma$, as desired.