A topological lower bound for the energy of a unit vector field on a closed Euclidean hypersurface

TL;DR

This paper establishes a topological lower bound for the energy of unit vector fields on closed Euclidean hypersurfaces, linking it to the degree of the Gauss map, and explores properties of certain functionals related to the immersion degree.

Contribution

It introduces a new lower bound for the energy of vector fields based on the Gauss map degree and studies the properties of functionals on hypersurfaces, including minimization by Hopf flows.

Findings

Lower bound for energy depends on Gauss map degree

Hopf flows minimize the functional  on spheres

Functionals  share properties related to immersion degree

Abstract

For a unit vector field on a closed immersed Euclidean hypersurface $M^{2n+1}$, $n\geq 1$, we exhibit a nontrivial lower bound for its energy which depends on the degree of the Gauss map of the immersion. When the hypersurface is the unit sphere $\mathbb{S}^{2n+1}$, immersed with degree one, this lower bound corresponds to a well established value from the literature. We introduce a list of functionals $\mathcal{B}_k$ on a compact Riemannian manifold $M^{m}$, $1\leq k\leq m$, and show that, when the underlying manifold is a closed hypersurface, these functionals possess similar properties regarding the degree of the immersion. In addition, we prove that Hopf flows minimize $\mathcal{B}_n$ on $\mathbb{S}^{2n+1}$.

Equations

$$E(\vec{v})=\frac{1}{2}{\int }_M\Vert \nabla \vec{v}{\Vert }^2+\frac{m}{2}\mathrm{vol}(M).$$

$$\mathcal{B}(\vec{v})=\frac{1}{(m-1)\mathrm{vol}({\mathbb{S}}^m)}{\int }_M\Vert \nabla \vec{v}{\Vert }^2,$$

$$E(\vec{v})=\frac{(m-1)\mathrm{vol}({\mathbb{S}}^m)}{2}\mathcal{B}(\vec{v})+\frac{m}{2}\mathrm{vol}(M).$$

$$\mathrm{vol}(\vec{v})={\int }_M\sqrt{\det (I+(\nabla \vec{v})(\nabla \vec{v}{)}^{\ast })},$$

$$\mathrm{vol}(\vec{v})-\mathrm{vol}(M)\ge \frac{\mathrm{vol}({\mathbb{S}}^{n+1})}{\mathcal{S}}|\mathrm{deg}(\nu )|,$$

$$E(\vec{v})\ge C(n)\frac{|\mathrm{deg}(\nu )|\mathrm{vol}({\mathbb{S}}^{2n+1})}{S^{[2n-1]}}+\frac{2n+1}{2}\mathrm{vol}(M^{2n+1})$$

$$\left(\frac{2n+1}{2}+\frac{n}{2n-1}\right)\mathrm{vol}({\mathbb{S}}^{2n+1}).$$

$$E(\vec{v})\ge \left(\frac{2n+1}{2}{r}^{2n+1}+\frac{n}{2n-1}{r}^{2n-1}\right)\mathrm{vol}({\mathbb{S}}^{2n+1}).$$

$${\mathcal{B}}_k(\vec{v})={\int }_M\underset{\text{k-times}}{\underbrace{\Vert \nabla \vec{v}\wedge \cdots \wedge \nabla \vec{v}{\Vert }^2}}.$$

$${\mathcal{B}}_n(\vec{v})\ge \left(\genfrac{}{}{0ex}{}{2n}{n}\right){\int }_M|{\sigma }_{2n}(\mathcal{V})|.$$

$${\mathcal{B}}_n(\vec{v})\ge \frac{|\mathrm{deg}(\nu )|}{\mathcal{S}}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)\mathrm{vol}({\mathbb{S}}^{2n+1}),$$

$$d\varphi^{\vec{v}}_{t}=\left(\begin{array}[]{ccc|c}\lx@intercol\hfil\hbox{\multirowsetup\raisebox{2.0pt}{$h_{ij}+ta_{ij}$}}\hfil\lx@intercol\vrule\lx@intercol&h_{2n+1\;1}+ta_{1\,2n+1}\\ &&&{\vdots}\\ &&&h_{2n+1\;2n}+ta_{2n\,2n+1}\\ \hline\cr\sqrt{1+t^{2}}h_{2n+1\;1}&\cdots&\sqrt{1+t^{2}}h_{2n+1\;2n}&\sqrt{1+t^{2}}h_{2n+1\;2n+1}\end{array}\right).$$

$$\det ((h_{AB})+t(a_{AB}))$$

$$\det (d{\phi }_t^{\vec{v}})=\sqrt{1+t^2}\sum _{k=0}^{2n}{\eta }_k{t}^k$$

$${\eta }_0$$

$${\eta }_1$$

$${\eta }_2$$

$${\eta }_{2n}$$

$${\int }_M{\eta }_k=\{\begin{array}{ll}\mathrm{deg}(\nu )\left(\genfrac{}{}{0ex}{}{n}{k/2}\right)\mathrm{vol}({\mathbb{S}}^{2n+1}),& \text{mbox}ifk\text{mbox}is\text{mbox}even,\\ 0,& \text{mbox}ifk\text{mbox}is\text{mbox}odd.\end{array}$$

$$S^{[A]}=\underset{1\le i_1,\dots ,i_A\le 2n+1;x\in M}{\sup }\{\Vert S(u_{i_1})\wedge \cdots \wedge S(u_{i_A}){\Vert }_{\infty }\},$$

$$E(\vec{v})$$

$${\int }_M{\eta }_2$$

$$\int_{M}\eta_{2}=\int_{M}\left(\sum_{i,j,k,l}\left|\begin{array}[]{cc}a_{ij}&a_{ik}\\ a_{lj}&a_{lk}\end{array}\right|\det(\widehat{H}^{ij}_{lk})+\sum_{i,j,l}\left|\begin{array}[]{cc}a_{ij}&a_{j\,2n+1}\\ a_{lj}&a_{l\,2n+1}\end{array}\right|\det(\widehat{H}^{i\,2n+1}_{l\,2n+1})\right),$$

$${\int }_M{\eta }_2$$

$$=$$

$${\int }_M{\eta }_2\le \frac{1}{2}{\int }_M\sum _{A<D,B<C}\left({\tilde{a}}_{AB}^2+{\tilde{a}}_{AC}^2+{\tilde{a}}_{DB}^2+{\tilde{a}}_{DC}^2\right)S^{[2n-1]}.$$

$${\int }_M{\eta }_2\le n{S}^{[2n-1]}{\int }_M\sum _{A,B}{\tilde{a}}_{AB}^2\le n{S}^{[2n-1]}{\int }_M\Vert \nabla \vec{v}{\Vert }^2.$$

$${\int }_{{\mathbb{S}}^{2n+1}(r)}{\eta }_2\le \frac{2n-1}{2}{S}^{[2n-1]}{\int }_{{\mathbb{S}}^{2n+1}(r)}\Vert \nabla \vec{v}{\Vert }^2.$$

$$E(\vec{v})\ge C(n)\frac{|\mathrm{deg}(\nu )|\mathrm{vol}({\mathbb{S}}^{2n+1})}{S^{[2n-1]}}+\frac{2n+1}{2}\mathrm{vol}(M^{2n+1}),$$

$$C(n)=\{\begin{array}{ll}\frac{n}{2n-1},& \text{if}{M}^{2n+1}={\mathbb{S}}^{2n+1}(r),\\ \frac{1}{2},& \text{otherwise}.\end{array}$$

$$|{\eta }_{2n}|$$

$${\mathcal{B}}_k(\vec{v})={\int }_M\sum _{\begin{array}{c}{A}_1<\cdots <A_k\\ B_1<\cdots <B_k\end{array}}{\det }^2\left(a_{B_1\cdots B_k}^{A_1\cdots A_k}\right).$$

Full text

Icaro Gonçalves was supported by CNPq 141113/2013-8, and is partially supported by a scholarship from the National Postdoctoral Program, PNPD-CAPES. Adriana V. Nicoli is supported by CAPES-PROEX 1745551

1. Introduction and statement of the main results

Let $M^{m}$ be a compact oriented Riemannian manifold, $m\geq 2$, and let $\nabla$ denote its Levi-Civita connection. The energy of a unit vector field on $M$ is defined as the energy of the map $\vec{v}:M\rightarrow T_{1}M$, where $T_{1}M$ denotes the unit tangent bundle equipped with the Sasaki metric, (see [10] and [11])

[TABLE]

In [10], Wiegmink defines the total bending functional, a quantitative measure for the extent to which a unit vector field fails to be parallel with respect to the Levi-Civita connection $\nabla$ of a Riemannian manifold $M$. Precisely,

[TABLE]

and the energy of $\vec{v}$ may be written in terms of this functional as

[TABLE]

An important question regarding these functionals is whether one can find unit vector fields such that the minimum of the above functional is attained. Brito [3] showed that Hopf flows are absolute minima of the functional $\mathcal{B}$ in $\mathbb{S}^{3}$:

Theorem 1** (Brito, [3]).**

Hopf vector fields are the unique vector fields on $\mathbb{S}^{3}$ to minimize $\mathcal{B}$.

Gluck and Ziller proved that Hopf flows are also the unit vector fields of minimum volume, with respect to the following definition of volume,

[TABLE]

where $I$ is the identity and $(\nabla\vec{v})^{*}$ represents adjoint operator.

Theorem 2** (Gluck and Ziller, [8]).**

The unit vector fields of minimum volume on $\mathbb{S}^{3}$ are precisely the Hopf vector fields, and no others.

On the other hand, Reznikov compared this functional to the topology of an Euclidean hypersurface. Let $M^{n+1}$ be a smooth closed oriented immersed hypersurface in $\mathbb{R}^{n+2}$, endowed with the induced metric, and let $\mathcal{S}=\sup_{x\in M}\|S_{x}\|=\sup_{x\in M}|\lambda_{i}(x)|$, where $S_{x}$ is the second fundamental operator in $T_{x}M$, and $\lambda_{i}(x)$ are the principal curvatures.

Theorem 3** (Reznikov, [9]).**

For any unit vector field $\vec{v}$ on $M$ we have

[TABLE]

where $\deg(\nu)$ is the degree of the Gauss map $\nu:M\rightarrow\mathbb{S}^{n+1}$.

In this short note, we take an odd dimensional hypersurface $M^{2n+1}$ and relate the energy of a unit vector field $\vec{v}$ to the topology of the immersion of $M$, by means of the degree of the Gauss map. Our main theorem reads

Theorem A**.**

For a unit vector field on a closed oriented Euclidean hypersurface $M^{2n+1}$,

[TABLE]

where $S^{[2n-1]}$ and $C(n)$ are constants depending on the immersion of $M$ and on $n$ (their precise definition will be given later).

Theorem A provides a topological obstruction to small values of the energy in a Riemannian manifold, specifically in a hypersurface in the Euclidean space. Two non-homotopic immersions will possess two different normal degrees; the bigger this value, the bigger the energy of a given unit vector field. As far as the authors know, this is the first connection between the topology of an immersion and the energy of a unit vector field.

A special case is the unit sphere $\mathbb{S}^{2n+1}$. Borreli et al [2] constructed a family of unit vector fields on $\mathbb{S}^{2n+1}$ with energy converging to the energy of a radial vector field. Its value turned out to be the infimum for the energy of unit vector fields without singularities.

Theorem 4** (Borreli, Brito and Gil-Medrano, [2]).**

The infimum of $E$ among all globally defined unit smooth vector fields of the sphere $\mathbb{S}^{2n+1}$ $(n\geq 2)$ is

[TABLE]

This value is not attained by any globally defined unit smooth vector field.

By a theorem of Hopf, $\deg(\nu)$ is the same for homotopic immersions of a compact hypersurface. Thus, it is interesting to look at spheres of different radii. Applying the procedure from theorem A to a sphere of radius $r$, we have

Corollary 1**.**

For $r>0$, let $\mathbb{S}^{2n+1}(r)$ be immersed in $\mathbb{R}^{2n+2}$ with normal degree one. Then

[TABLE]

Consequently, when $r=1$ we recover the value from theorem 4. This inequality is well known from the literature, see [6] for a further discussion on the energy of vector fields possessing isolated singularities, as well as a proof of a general inequality regarding ${\rm Ricci}(\vec{v},\vec{v})$ on a given compact Riemanninan manifold.

We also discuss a list of functionals having properties similar to the total bending of flows, and determine a lower value for each one of them depending again on $\deg(\nu)$. Let $\vec{v}$ be a unit vector field on a compact oriented Riemannian manifold $M^{m}$. For every $1\leq k\leq m-1$, define

[TABLE]

If $\sigma_{2n}$ denotes the $2n$-th elementary symmetric function, and $\mathcal{V}$ is the restriction of $\nabla\vec{v}$ to $V^{\perp}$ then our last theorem reads

Theorem B**.**

Let $M^{2n+1}$ be a compact oriented Riemannian manifold, and let $\vec{v}$ be a unit vector field on $M$. Then

[TABLE]

Furthermore, when $M^{2n+1}$ is a closed Euclidean hypersurface,

[TABLE]

where $\mathcal{S}$ is the aforementioned constant.

As a consequence, we deduce the following

Corollary 2**.**

Hopf vector fields minimize $\mathcal{B}_{n}$ on $S^{2n+1}$.

Hopf vector fields on $\mathbb{S}^{3}$ are absolute minima of the energy, [3], and on $\mathbb{S}^{2n+1}$, for $n\geq 2$, they are critical but unstable points of the energy functional, [11]. Despite these properties on higher dimensional spheres, the functionals $\mathcal{B}_{n}$ on $S^{2n+1}$ are an attempt to provide a list of functionals that are minimized by Hopf vector fields, having similar features when compared to the energy and/or total bending. This result should also be compared to a mean curvature correction of the total bending provided by the first author on [3].

2. Curvature integrals for a closed hypersurface

The proofs of theorem A, corollary 1 and of 5 rely on a list of curvature integrals, described in this Section.

We may assume that $M^{2n+1}$ is oriented, so the normal map $\nu:M^{2n+1}\rightarrow\mathbb{S}^{2n+1}$, $\nu(x)=N(x)$, is well defined, where $N$ is a unitary normal field. Let $\langle\cdot,\cdot\rangle$ be the induced Riemannian metric on $M$. Let $\vec{v}:M\rightarrow TM$ be a smooth unit vector field on $M$, and take the orthonormal basis $\{e_{1},\dots,e_{2n},e_{2n+1}:=\vec{v}\}$ at each point $x\in M$. We fix some notation: for $1\leq A,B\leq 2n+1$, set $h_{AB}=\langle S(e_{A}),e_{B}\rangle$ and $a_{AB}=\langle\nabla_{e_{B}}\vec{v},e_{A}\rangle$; it follows that $a_{2n+1\,B}=\langle\nabla_{e_{B}}\vec{v},\vec{v}\rangle=0$, for all $B$. For a real number $t>0$, define $\varphi^{\vec{v}}_{t}:M^{2n+1}\to\mathbb{S}^{2n+1}(\sqrt{1+t^{2}})$, by $\varphi^{\vec{v}}_{t}(x)=\nu(x)+t\vec{v}(x)$. With respect to the aforementioned basis of $T_{x}M$ and setting $\left\{e_{1},\dots,e_{2n},\frac{\vec{v}}{\sqrt{1+t^{2}}}-t\frac{N}{\sqrt{1+t^{2}}}\right\}$ as an orthonormal basis of $T_{\varphi^{\vec{v}}_{t}(x)}\mathbb{S}^{2n+1}(\sqrt{1+t^{2}})$, we have that

[TABLE]

Multilinearity of determinant simplifies computations concerning an explicit formula for $\det(d\varphi^{\vec{v}}_{t})$ written in terms of the second fundamental form of $M$ and components depending on the normal bundle of $\vec{v}$.

[TABLE]

Therefore,

[TABLE]

where

[TABLE]

The fact that $\det((h_{AB})+t(a_{AB}))=\sum_{k}\eta_{k}t^{k}$ shows that, for $1\leq k\leq 2n$, $\eta_{k}$ does not depend on the choice of basis.

Since $\varphi^{\vec{v}}_{t}$ and $\nu$ are homotopic, we have that $\deg(\varphi^{\vec{v}}_{t})=\deg(\nu)$. On the other hand, if $f:X\to Y$ is a smooth map between two manifolds of the same dimension, say $m$, then for any $m$-form $\omega$ on $Y$, the degree formula reads $\int_{X}f^{*}(\omega)=\deg(f)\int_{Y}\omega$. By change of variables we conclude that

[TABLE]

These invariants have been computed and applied to questions concerning geometry of foliations on hypersurfaces, [5].

3. Proof of theorem A

We start our approach by defining a list of numbers regarding wedge products of the shape operator of $M$ and their restriction to a list of vectors on a point.

Definition 1**.**

If $\{u_{1},\dots,u_{2n+1}\}$ is an orthonormal basis at $x\in M$, then, for each $1\leq A\leq 2n+1$,

[TABLE]

where $\|\cdot\|_{\infty}$ denotes the maximum norm, naturally extended to $\Lambda^{A}(M)$.

We notice that $S^{[A]}$ is well defined since $M$ is a compact hypersurface. With respect to $\{e_{1},\dots,e_{2n},\vec{v}\}$, the energy functional is written as

[TABLE]

this means that $\eta_{2}$ is the natural choice among all $\eta_{k}$ in order to determine a lower value for $E(\vec{v})$. From the last Section, we have

[TABLE]

By the definition of the matrix $(a_{AB})$ and the fact that two of its entries appear in the above summation, we are able to display all $2\times 2$ minors of $(a_{AB})$, times a minor depending on the second fundamental form matrix of $M$. Precisely,

[TABLE]

where $\widehat{H}^{AB}_{CD}$ is a $(2n-1)\times(2n-1)$ matrix which comes from $(h_{AB})$ by removing the $A$-th and $C$-th lines, and $B$-th and $D$-th columns.

For $0\leq k\leq 2n$ the functions $\eta_{k}$ are invariant under any change of basis. We may assume that $\eta_{2}$ is computed with respect to a basis that diagonalizes the second fundamental form matrix of $M$. In this new basis, we have a matrix $(\widetilde{a}_{AB})$ and its last line might be different from zero. So we write

[TABLE]

From our definition 1, $\prod_{F\neq A,C}h_{FF}\leq S^{[2n-1]}$. Employing the inequality \left|\begin{array}[]{cc}a&b\\ c&d\end{array}\right|\leq\frac{1}{2}(a^{2}+b^{2}+c^{2}+d^{2}), we have

[TABLE]

Now comes the crucial distinction between theorem A and corollary 1; that is, between the constant $C(n)$ obtained for an arbitrary hypersurface and by restricting to a sphere of radius $r$. For an arbitrary closed hypersurface $M^{2n+1}$, we are able to diagonalize $(h_{AB})$ but we can not control which entries in $(\widetilde{a}_{AB})$ will remain different from zero. This implies that when we count its squared entries in the last inequality we end up with $2n$ elements. Hence

[TABLE]

On the other hand, if $M=\mathbb{S}^{2n+1}(r)$, then $(h_{AB})$ is $1/r$ times the identity matrix, $(h_{AB})=(1/r)\cdot I$. This means that for any choice of basis $(h_{AB})$ is a diagonal matrix. In particular, we can employ the orthonormal basis $\{e_{1},\dots,e_{2n},e_{2n+1}:=\vec{v}\}$, and $(a_{AB})$ is the same as before, i.e. $(a_{2n+1\ B})=\langle\nabla_{e_{B}}\vec{v},\vec{v}\rangle=0$ for all $1\leq B\leq 2n+1$. In this case, when we count the terms in $\sum_{A<D,B<C}\left(a_{AB}^{2}+a_{AC}^{2}+a_{DB}^{2}+a_{DC}^{2}\right)$ we have a smaller number, $2n-1$. Thus

[TABLE]

Therefore,

[TABLE]

where

[TABLE]

Notice that when $M^{2n+1}=\mathbb{S}^{2n+1}(r)$, its volume is $r^{2n+1}$ times the volume of $\mathbb{S}^{2n+1}$. In addition, $S^{[2n-1]}=r^{-2n+1}$, because $(h_{AB})=(1/r)\cdot I$. This concludes the proof of corollary 1.

4. Known results and a list of new functionals

This section is intended to show some direct consequences of equation 6, as well as proving theorem B. We start by considering the integral of the last invariant $\eta_{2n}$ and then we obtain the main ingredient in the proof of theorem 3 from [9]. Finally, we discuss the list of “higher order” total bending functionals on Riemannian manifolds, and study their lower bounds on closed Euclidean hypersurfaces.

4.1. Another view on theorem 3

By definition, the last line of $(a_{AB})$ is zero ($\vec{v}$ is unitary), so we let $m_{A}$, $1\leq A\leq 2n+1$, denote its $2n$-minors. Thus, $\eta_{2n}=\sum_{A}(-1)^{A+1}h_{2n+1A}m_{A}$, which implies

[TABLE]

For the last inequality, see for example Lemma 1 in [9]. The proof of the theorem 3 is finished by integration over $M$.

4.2. Higher order functionals: proof of theorem B

The fact that $\vec{v}$ is unitary implies that $\mathcal{B}_{m}(\vec{v})=0$, and $\mathcal{B}_{1}(\vec{v})$ is, up to a constant, the total bending of $\vec{v}$. All $\mathcal{B}$ functionals can be written as integrals of functions of $k$-minors from the matrix $(a_{AB})$,

[TABLE]

Assume that $M$ has dimension $2n+1$. When $1\leq i,j\leq 2n$, the matrix $(a_{ij})=(\langle\nabla_{e_{j}}\vec{v},e_{i}\rangle)$ describes the behavior of the distribution normal to $\vec{v}$. We are going to compare its determinant to the integrand in the functional $\mathcal{B}_{n}$.

If we omit the $n$-minors having at least one element of the type $a_{i\ 2n+1}=\langle\nabla_{\vec{v}}\vec{v},e_{i}\rangle$, then

[TABLE]

We follow Sections 3 and 4 from [4], and Chapter IV of [7]. In general, the distribution normal to $\vec{v}$ is not integrable. Even though $(a_{ij})$ is not symmetric, we can find a local basis in which $(a_{ij})$ has the form of a upper triangular matrix,

[TABLE]

Let $D=\text{diag}(|\lambda_{1}|,\dots,|\lambda_{r}|,\sqrt{x_{1}^{2}+y_{1}^{2}},\sqrt{x_{1}^{2}+y_{1}^{2}},\dots,\sqrt{x_{s}^{2}+y_{s}^{2}},\sqrt{x_{s}^{2}+y_{s}^{2}})$ be a $2n\times 2n$ diagonal matrix. By construction, $\det D\geq|\det(a_{ij})|$. On the other hand,

[TABLE]

simply because most elements above the main diagonal in $(a_{ij})$ are different from zero, which is the case for $D$.

The main result of Section 3 in [4] is the “Fundamental Lemma”, which states an inequality between the volume of a $2n\times 2n$ diagonal matrix and the sum of its even elementary symmetric functions. In proving this lemma, the authors deduced the following inequality (cf. second inequality on page 307 of [4]; we also refer to [7], equations (IV.16) and (IV.21) pages 55 and 56, respectively)

[TABLE]

Since $\sigma_{2n}(\mathcal{V})=\det(a_{ij})$, this completes the proof of 4.

The simplest invariant depending on $\vec{v}$ in Section 2 is $\eta_{2n}$. If $\mathcal{S}$ is the number defined in theorem 3, then both of them combines to establish 5.

4.3. Hopf flows on $\mathbb{S}^{2n+1}$

We proceed to prove Corollary 2. Hopf vector fields are tangent to the fibers of $\mathbb{S}^{1}\hookrightarrow\mathbb{S}^{2n+1}\to\mathbb{C}{\rm P}^{n}$, and the matrix associated to the second fundamental form of a given $H$ is a $2n\times 2n$ matrix of the type

[TABLE]

having $n$ blocks of the form \left(\begin{array}[]{cc}0&-1\\ 1&0\\ \end{array}\right) in a diagonal and zeros everywhere else. In computing $\mathcal{B}_{n}(H)$ on $\mathbb{S}^{2n+1}$, we need to count how many $n$-minors different from zero the above matrix has. A given $n\times n$ sub-matrix is obtained by removing $n$ rows and $n$ columns of $(a_{ij})$. At the end of this process, all rows and columns of this $n\times n$ sub-matrix have exactly one element, $\pm 1$. By rearranging the rows in this sub-matrix we get a $n\times n$ diagonal matrix, and this matrix has determinant $\pm 1$. Thus, we have ${2n\choose n}$ non-zero $n$-minors, each one evaluating $\pm 1$. Therefore, $\mathcal{B}_{n}(H)={2n\choose n}{\rm vol}(\mathbb{S}^{2n+1})$.

Acknowledgment

We thank an anonymous referee for valuable suggestions and a critical look at a previous version of this manuscript.