A DDVV Conjecture for Riemannian Maps

Abstract

The Wintgen inequality is a significant result in the field of differential geometry, specifically related to the study of submanifolds in Riemannian manifolds. It was discovered by Pierre Wintgen. In the present work, we deal with the Riemannian maps between Riemannian manifolds that serve as a superb method for comparing the geometric structures of the source and target manifolds. This article is the first to explore a well-known conjecture, called DDVV inequality (a conjecture for Wintgen inequality on Riemannian submanifolds in real space forms proven by P.J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken), for Riemannian maps, where we consider different space forms as target manifolds. There are numerous research problems related to such inequality in various ambient manifolds. These problems can all be explored within the general framework of Riemannian maps between various Riemannian manifolds equipped with notable geometric structures.

1. Introduction

Riemannian maps, first presented by A.E. Fischer [1] in 1992, are specific types of mappings between Riemannian manifolds that preserve certain geometric structures or properties. These maps play a crucial role in differential geometry by facilitating the comparison and analysis of geometric features across different manifolds. These maps extend the familiar concepts of Riemannian submersions and isometric immersions. Riemannian maps are used in various mathematical and applied contexts to understand the relationships between different geometric spaces, analyze geometric structures, and solve problems in physics, computer science, and beyond, where geometric structures play a key role.

Let $(B,g_B)$ and $(F,g_F)$ be two Riemannian manifolds with $dim(B)=r$ and $dim(F)=s$. Suppose there is a smooth mapping $P:(B,g_B)\to (F,g_F)$, where $0\le rank(P)=p\le min\{r,s\}$. At any point $x\in B$, we define the kernel space of $P_{*}$ by $V_x=ker{(P_{*})}_x$ and the orthogonal complementary space to $ker{(P_{*})}_x$ by $H_x={(ker{(P_{*})}_x)}^{\perp }$ (horizontal space). Consequently, the tangent space $T_{x}B$ at x of B can be expressed as the direct sum of these two spaces:

$$\begin{array}{c}\hfill T_{x}B=V_x\oplus H_x.\end{array}$$

(1)

Consider the range of $P_{*}$, denoted by $range{(P_{*})}_x$, at point x of B and the orthogonal complementary space to $range{(P_{*})}_x$ by ${(range{(P_{*})}_x)}^{\perp }$ in the tangent space of F at x, denoted by $T_{P(x)}F$. Since $p\le min\{r,s\}$, we have ${(range(P_{*}))}^{\perp }\ne 0$. Hence, $T_{P(x)}F$ can be decomposed as follows:

$$\begin{array}{c}\hfill T_{P(x)}F=range{(P_{*})}_x\oplus {(range{(P_{*})}_x)}^{\perp }.\end{array}$$

(2)

Definition 1. 

A smooth map $P:(B,g_B)\to (F,g_F)$ with $0\le rank(P)=p\le min\{r,s\}$ is said to be a Riemannian map at $y\in B$ if the horizontal restriction

$$\begin{array}{c}\hfill P_{*y}^h:{(ker{(P_{*})}_y)}^{\perp }\to range{(P_{*})}_y\end{array}$$

(3)

is a linear isometry, that is,$P_{*}$satisfies the equation

$$\begin{array}{c}\hfill g_F(P_{*}\hat{X},P_{*}\hat{Y})=g_B(\hat{X},\hat{Y}),\end{array}$$

(4)

for any vector fields$\hat{X}$and$\hat{Y}$tangent to H. Furthermore, if P acts as a Riemannian map at each point$y\in B$, then P is known as a Riemannian map (shortly, RM).

Remark 1. 

It easy to observe that RMs are isometric immersions if $ker(P_{*})=\left\{0\right\}$ and that they are Riemannian submersions if ${(range(P_{*}))}^{\perp }=\left\{0\right\}$.

Remark 2. 

A notable property of RMs was shown by Fischer [1], who stated that they satisfy the generalized eikonal equation, which is a bridge between geometric optics and physical optics.

In summary, Riemannian maps are versatile tools that facilitate the transfer and analysis of geometric, topological, and physical properties across different spaces, finding applications in a wide range of scientific and engineering disciplines: RMs can be used to understand the deformation and mapping of spacetime in the context of general relativity; they are also used to study shapes and their transformations in computer graphics and computer vision and help us to understand the relationships between different geometric structures, as well as having many more uses.

On the other hand, the renowned embedding theorem devised by Nash provided significant inspiration for defining precise relationships between the intrinsic and extrinsic invariants of Riemannian submanifolds. The derivation of optimal inequalities for submanifolds in diverse ambient spaces is primarily influenced by isometric immersions. The Wintgen inequality is a fundamental result in the differential geometry of submanifolds, particularly in the study of surfaces in higher-dimensional spaces. It outlines the relationship between the intrinsic and extrinsic geometry of a submanifold. In [2], P. Wintgen demonstrated that for any surface in ${\mathbb{E}}^4$, the Gauss curvature K, the squared mean curvature ${\left|\right|\mathcal{H}\left|\right|}^2$, and the normal curvature $K^{\perp }$ always satisfy the following inequality:

$$\begin{array}{c}\hfill K\le {\left|\right|\mathcal{H}\left|\right|}^2-\left|K^{\perp }\right|.\end{array}$$

(5)

The equality holds if and only if the ellipse of the curvature of a surface in ${\mathbb{E}}^4$ is a circle.

Geometric Interpretation: The Wintgen inequality expresses a balance between the intrinsic curvature (the Gaussian curvature) and the extrinsic curvature (the mean curvature and the traceless part of the second fundamental form) of the surface. It essentially states that the intrinsic curvature cannot be too large compared to the extrinsic curvature measures.

Example 1. 

Consider a minimal surface in ${\mathbb{E}}^3$, where the mean curvature $\mathcal{H}=0$. The Wintgen inequality simplifies to the following:

$$K\le -|K^{\perp }|.$$

In this case, the Gaussian curvature K is non-positive, reflecting the fact that minimal surfaces tend to have saddle points.

P.J. De Smet, F. Dillen, L. Verstraelen, and L. Vrancken [3] proposed the DDVV conjecture, often known as the generalized Wintgen inequality, the DDVV inequality, or the normal scalar curvature conjecture, for a submanifold $M^n$ of a real space form ${\overline{M}}^{n+m}(c)$ of constant sectional curvature c. The conjecture is expressed as follows:

$$\begin{array}{c}\hfill {\left|\right|\mathcal{H}\left|\right|}^2\ge {\rho }^{nor}+{\rho }^{\perp }-c,\end{array}$$

(6)

where ${\rho }^{nor}$ represents the (intrinsic) normalized scalar, ${\rho }^{\perp }$ denotes the (extrinsic) normalized normal scalar, and ${\left|\right|\mathcal{H}\left|\right|}^2$ represents the (extrinsic) squared mean curvatures of M. It is worth mentioning that number of geometers achieved partial results concerning the DDVV conjecture (for example, [4]). However, the conjecture, in its general form, was independently proven in [5] and in [6].

The DDVV inequality, initially applied to submanifolds in real space forms, has seen significant extensions and applications across a variety of geometric contexts thanks to the efforts of many researchers: It was notably extended by I. Mihai for Lagrangian and slant submanifolds in complex space forms [7], alongside extensions to Legendrian submanifolds in Sasakian space forms [8]. M.E. Aydin, A. Mihai, and I. Mihai [9] further broadened its application to statistical submanifolds in statistical manifolds of constant curvature. J. Roth [10] developed a version of the DDVV inequality for submanifolds of warped products in the form $I{\times }_f{M}^n(c)$, where I is an interval and a smooth function $f:I\to \mathbb{R}$ that never vanishes. C. Murathan and B. Sahin [11] sourced inspiration from Roth and proved a general Wintgen inequality for the statistical submanifolds of statistical warped product manifolds. The generalized Wintgen inequality for a Legendrian submanifold in standard warped product manifolds, specifically in $\beta$-Kenmotsu manifolds, was derived by A.N. Siddiqui, et al. [12]. In advancing this topic, the generalized Wintgen inequality is established for statistical submanifolds in Hessian manifolds of constant Hessian curvature [13]. Indeed, a number of geometers have successfully derived the Wintgen inequality in different settings, and these works have contributed to the deepening of our understanding of curvature relations and geometric inequalities in these specialized contexts.

Therefore, the main aim of this article is to extend this conjecture for an RM with a real space form as a target manifold. Also, we discuss the same for Riemannian manifold of quasi-constant curvature (and of nearly quasi-constant curvature) as a target manifold. Moreover, we extend this inequality to the case of Riemannian maps in which the target manifold is a complex space form.

2. Preliminaries

For a Riemannian map (shortly, RM) $P:(B,g_B)\to (F,g_F),$ its differential, denoted $P_{*},$ acts as a section of the bundle $Hom(TB,P^{-1}TF)$ on B. Here, $P^{-1}TF$ represents the pullback bundle, with fiber $P^{-1}{T}_{x}B=T_{P(x)}F$ for each point $x\in B$. The bundle $Hom(TB,P^{-1}TF)$ has an induced connection D from the Levi-Civita connection $D^B$ on B. Then, the second fundamental form of P can be expressed as [14]

$$\begin{array}{c}\hfill (D{P}_{*})(\hat{X},\hat{Y})=D^P{P}_{*}(\hat{Y})-P_{*}(D_{\hat{X}}^B\hat{Y}),\end{array}$$

(7)

for any $\hat{X},\hat{Y}\in \Gamma (TB)$, where $D^P$ refers to the pullback connection. This second fundamental form is known to be symmetric. Additionally, it has been established that $(D{P}_{*})(\hat{X},\hat{Y})$ has no components in $range(P_{*})$, provided that $\hat{X},\hat{Y}\in \Gamma ({(ker(P_{*}))}^{\perp })$, where ${(ker(P_{*}))}^{\perp }$ denotes the subbundle of $P^{-1}(TF)$ with fiber ${(P_{*}(T_{x}B))}^{\perp }$, $x\in B$.

The Levi-Civita connection on F is denoted by $D^F$ and its pullback along P. The Weingarten formula for P is given by [14]

$$\begin{array}{c}\hfill D_{\hat{X}}^{P}N=S_N{P}_{*}\hat{X}+D_{\hat{X}}^{P\perp }N,\end{array}$$

(8)

where $D_{\hat{X}}^{P\perp }N$ is the orthogonal projection of $D_{\hat{X}}^{P}N$, $\hat{X}\in \Gamma (TB)$ and $N\in \Gamma ({(range(P_{*}))}^{\perp })$, and $S_N{P}_{*}\hat{X}$ is the shape operator defined on $P_{*}(TB)$. The second fundamental form is bilinear, and through direct calculation, it is shown that [14]

$$\begin{array}{c}\hfill g_F(S_N{P}_{*}\hat{X},S_N{P}_{*}\hat{Y})=g_F(N,(D{P}_{*})(\hat{X},\hat{Y})),\end{array}$$

(9)

for all $\hat{X},\hat{Y}\in \Gamma ({(ker(P_{*}))}^{\perp })$ and $N\in \Gamma ({(range(P_{*}))}^{\perp })$. Given that $(D{P}_{*})$ is symmetric, this implies that $S_N$ is a symmetric linear transformation of $range(P_{*})$.

Consider $R^B$ and $R^F$ as the curvature tensor fields of $D^B$ and $D^F$, respectively. Consequently, the Gauss equation can be expressed as follows [14]:

$$\begin{array}{ccc}\hfill g_F(R^F(P_{*}\hat{X},P_{*}\hat{Y})P_{*}\hat{Z},P_{*}\hat{W})& =& g_B(R^B(\hat{X},\hat{Y})\hat{Z},\hat{W})+g_F((D{P}_{*})(\hat{X},\hat{Z}),(D{P}_{*})(\hat{Y},\hat{W}))\hfill \\ & & -g_F((D{P}_{*})(\hat{X},\hat{W}),(D{P}_{*})(\hat{Y},\hat{Z})),\hfill \end{array}$$

(10)

for $\hat{X},\hat{Y},\hat{Z},\hat{W}\in \Gamma ({(ker(P_{*}))}^{\perp })$.

The Ricci equation is expressed as [14]

$$\begin{array}{ccc}\hfill g_F(R^F(P_{*}\hat{X},P_{*}\hat{Y})N,V)& =& g_F(R^{F\perp }(P_{*}\hat{X},P_{*}\hat{Y})N,V)\hfill \\ & & +g_F([S_N,S_V]P_{*}\hat{X},P_{*}\hat{Y}),\hfill \end{array}$$

(11)

for $N,V\in \Gamma ({(range(P_{*}))}^{\perp })$. Here, $[S_N,S_V]=S_N{S}_V-S_V{S}_N$.

For any $x\in B$, consider $\{v_1,\cdots ,v_p,v_{p+1},\cdots ,v_s\}$ to be an orthonormal basis for $T_{x}B$, where $\{v_1,\cdots ,v_p\}$ forms an orthonormal basis for $H_x$ and $\{v_{p+1},\cdots ,v_s\}$ forms an orthonormal basis for $ker(P_{*x})$. The scalar curvature, denoted as $scal$, on $H_x$ is calculated by using

$$\begin{array}{c}\hfill scal=\sum _{1\le i,j\le p}{g}_B(R^B(v_i,v_j)v_j,v_i)\end{array}$$

(12)

and the normalized scalar curvature of $H_x$ is defined by

$$\begin{array}{c}\hfill {\rho }^{nor}={\displaystyle \frac{2scal}{p(p-1)}}.\end{array}$$

(13)

A new notion of normal scalar curvature, denoted as ${\rho }^{\perp }$, is defined for every point $x\in B$ as follows:

$$\begin{array}{c}\hfill {\rho }^{\perp }={\displaystyle \frac{1}{p(p-1)}}{\left(\sum _{1\le i<j\le p}\sum _{1\le a<b\le s-p}{g}_F(R^{F\perp }(v_i,v_j)V_a,V_b)\right)}^{1/2},\end{array}$$

(14)

where $\{V_{p+1},V_{p+2},V_{p+3},\cdots ,V_s\}$ represents an orthonormal basis of ${(range(P_{*x}))}^{\perp }$.

3. Riemannian Maps to Real Space Forms

Consider the target manifold as a real space form, which is an s-dimensional Riemannian (shortly, Rie.) manifold $(F,g_F)$ with constant sectional curvature c, denoted by $F(c)$. In this context, $R^F$ of $F(c)$ is expressed as

$$\begin{array}{c}\hfill g_F(R^F(\hat{X},\hat{Y})\hat{Z},\hat{W})=c\{g_F(\hat{X},\hat{W})g_F(\hat{Y},\hat{Z})-g_F(\hat{X},\hat{Z})g_F(\hat{Y},\hat{W})\},\end{array}$$

(15)

for the vector fields $\hat{X},\hat{Y},\hat{Z},\hat{W}$ on F.

We prove the following result by utilizing Lemma 2.1 from [3], which states that

Lemma 1. 

Let $a_i\in \mathbb{R}$ for $i=\{1,2,3,\cdots ,n\}$. Then

$$\sum _{i<j}{(a_i-a_j)}^2\ge {\displaystyle \frac{n}{2}}{(a_1-a_2)}^2$$

and equality holds if and only if

$${\displaystyle \frac{1}{2}}(a_1+a_2)=a_3=a_4=\cdots =a_n.$$

If

$$\sum _{i<j}{(a_i-a_j)}^2={\displaystyle \frac{n}{2}}{(a_1-a_2)}^2={\displaystyle \frac{n}{2}}{(a_k-a_l)}^2,$$

then

$$a_1=a_2=a_3=a_4=\cdots =a_n,$$

for integers k and l such that $1\le k<l\le n$ and $(k,l)\ne (1,2)$.

Theorem 1. 

Let P be an RM from an r-dimensional Rie. manifold $(B,g_B)$ onto a $p+two$-dimensional real space form $(F(c),g_F)$ with $1<rank(P)=p<p+2$. Then, we have

$$\begin{array}{c}\hfill {\left|\right|trace(h)\left|\right|}^2\ge p^2({\rho }^{\perp }+{\rho }^{nor}-c).\end{array}$$

Proof. 

We choose an orthonormal basis of $H_x$ of $\left\{v_i\right|i=1,2,3,\cdots ,p\}$ and choose $V_2$ orthogonal to $V_1$, which is in the direction of the mean curvature vector. Then, we compute (15) by inputting $\hat{X}=v_i$, $\hat{Y}=v_j$, $\hat{W}=V_1$, $\hat{Z}=V_2$ as follows:

$$\begin{array}{c}\hfill g_F(R^F(v_i,v_j)V_1,V_2)=0.\end{array}$$

Thus, (11) reduces to

$$\begin{array}{ccc}\hfill g_F(R^{F\perp }(v_i,v_j)V_1,V_2)& =& -g_F([S_{V_1},S_{V_2}]v_i,v_j)\hfill \\ & =& g_F(S_{V_2}{v}_i,S_{V_1}{v}_j)-g_F(S_{V_1}{v}_i,S_{V_2}{v}_j)\hfill \\ & =& (h_{ii}^2-h_{jj}^2)h_{ij}^1,\hfill \end{array}$$

where we use (9) and the notation $h_{ij}^c=g_F((D{P}_{*})(v_i,v_j),V_c)$. So

$$\begin{array}{c}\hfill {\displaystyle \frac{p(p-1)}{2}}{\rho }^{\perp }=\sqrt{\sum _{i<j}({(h_{ii}^2-h_{jj}^2)}^2{(h_{ij}^1)}^2)}.\end{array}$$

Then, we can compute the following:

$$\begin{array}{ccc}\hfill {\left|\right|trace(h)\left|\right|}^2& =& g_F(\sum _{i=1}^p((D{P}_{*}))(v_i,v_i),\sum _{i=1}^p((D{P}_{*}))(v_i,v_i))\hfill \\ & =& g_F(trace(h),trace(h))\hfill \\ & =& {(\sum _{i=1}^p{h}_{ii}^1)}^2+{(\sum _{i=1}^p{h}_{ii}^2)}^2\hfill \\ & =& {\displaystyle \frac{1}{p-1}}(\sum _{i<j}{(h_{ii}^1-h_{jj}^1)}^2+\sum _{i<j}{(h_{ii}^2-h_{jj}^2)}^2)\hfill \\ & & +({\displaystyle \frac{2p}{p-1}})\sum _{i<j}(h_{ii}^1{h}_{jj}^1+h_{ii}^2{h}_{jj}^2).\hfill \end{array}$$

(16)

Therefore, by employing (10) and taking into account our selected basis, we conclude that

$$\begin{array}{ccc}\hfill (p-1)(|\left|trace(h)\right|{|}^2-p^2{\rho }^{nor}+p^{2}c)& =& \sum _{i<j}{(h_{ii}^1-h_{jj}^1)}^2+\sum _{i<j}{(h_{ii}^2-h_{jj}^2)}^2)\hfill \\ & & +2p\sum _{i<j}{(h_{ij}^1)}^2.\hfill \end{array}$$

(17)

To introduce inequality into the above formula, we derive

$$\begin{array}{ccc}\hfill (p-1)(|\left|trace(h)\right|{|}^2-p^2{\rho }^{nor}+p^{2}c)& \ge & \sum _{i<j}({(h_{ii}^2-h_{jj}^2)}^2+2p{(h_{ij}^1)}^2).\hfill \end{array}$$

Lemma 1 and the relation ${(A+B)}^2\ge 4AB$ yield

$$\begin{array}{ccc}\hfill (p-1)(|\left|trace(h)\right|{|}^2-p^2{\rho }^{nor}+p^{2}c)& \ge & \sqrt{8p\sum _{k<l}\sum _{i<j}({(h_{ii}^2-h_{jj}^2)}^2{(h_{kl}^1)}^2)}\hfill \\ & \ge & 2p\sqrt{\sum _{k<l}({(h_{kk}^2-h_{ll}^2)}^2{(h_{kl}^1)}^2)}.\hfill \end{array}$$

Applying (11) and (14) to the above inequality, we arrive at

$$\begin{array}{c}\hfill {(p-1)(|\left|trace(h)\right||}^2-p^2{\rho }^{nor}+p^{2}c)\ge p^2(p-1){\rho }^{\perp }\end{array}$$

from which we deduce (2). □

In light of the aforementioned main theorem, we can say the following:

Theorem 2. 

Let P be an RM from an r-dimensional Rie. manifold $(B,g_B)$ onto an s-dimensional real space form $(F(c),g_F)$ with $1<rank(P)=p<s$. Then, we have

$$\begin{array}{c}\hfill {\left|\right|trace(h)\left|\right|}^2\ge p^2({\rho }^{\perp }+{\rho }^{nor}-c).\end{array}$$

Proof. 

We choose orthonormal bases of $H_x$ and ${(range(P_{*x}))}^{\perp }$ of $\left\{v_i\right|i=1,2,3,\cdots ,p\}$ and $\left\{V_a\right|a=p+1,p+2,p+3,\cdots ,s\}$, respectively. Then, by the Gauss equation, we have

$$\begin{array}{c}\hfill scal={\displaystyle \frac{p(p-1)}{2}}c+\sum _{a=1}^s\sum _{1\le i<j\le p}[h_{ii}^a{h}_{jj}^a-{(h_{ij})}^a],\end{array}$$

(18)

Next, we write ${\left|\right|trace(h)\left|\right|}^2$ as

$$\begin{array}{c}\hfill {(p-1)\left|\right|trace(h)\left|\right|}^2=\sum _a\sum _{i<j}(h_{ii}^a-h_{jj}^a)+2p\sum _a\sum _{i<j}{h}_{ii}^a{h}_{jj}^a,\end{array}$$

(19)

and use the following inequality from [5]:

$$\begin{array}{ccc}\hfill & & \sum _{a=1}^s\sum _{1\le i<j\le p}{(h_{ii}^a-h_{jj}^a)}^2+2p\sum _{a=1}^s\sum _{1\le i<j\le p}(h_{ij}^a)\hfill \\ & \ge & 2p{\left[\sum _{1\le a<b\le s}\sum _{1\le i<j\le p}{\left(\sum _{k=1}^p(h_{jk}^a{h}_{ik}^b-h_{ik}^a{h}_{jk}^b)\right)}^2\right]}^{1/2}.\hfill \end{array}$$

(20)

Then, from (14), (19), and inequality (20), and hence together with (18), we finally arrive at

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{p}}{\left|\right|trace(h)\left|\right|}^2-p{\rho }^{\perp }& \ge & {\displaystyle \frac{2}{p-1}}\sum _{a=1}^s\sum _{1\le i<j\le p}[h_{ii}^a{h}_{jj}^a-{(h_{ij})}^a]\hfill \\ & =& {\displaystyle \frac{2}{p-1}}(scal-{\displaystyle \frac{p(p-1)}{2}}c),\hfill \end{array}$$

which further gives

$$\begin{array}{ccc}\hfill {\left|\right|trace(h)\left|\right|}^2-p^2{\rho }^{\perp }& \ge & {\displaystyle \frac{2p^2}{p(p-1)}}(scal-{\displaystyle \frac{p(p-1)}{2}}c)\hfill \\ & =& p^2({\rho }^{nor}-c).\hfill \end{array}$$

□

To support Theorem 2 and (1), let us revisit several examples, as outlined below:

Example 2. 

We recall Example 1 from [15]. Consider the standard Rie. submersion $P:{\mathbb{S}}^{15}\to {\mathbb{S}}^8({\displaystyle \frac{1}{2}})$ between the 15-dimensional sphere ${\mathbb{S}}^{15}$ of constant sectional curvature 1 and the 8-dimensional sphere ${\mathbb{S}}^8({\displaystyle \frac{1}{2}})$ of constant sectional curvature 4 and radius ${\displaystyle \frac{1}{2}}$, with totally geodesic fibers ${\mathbb{S}}^7$ and the totally umbilical isometric immersion $\iota :{\mathbb{S}}^8({\displaystyle \frac{1}{2}})\to {\mathbb{R}}^9$. Then, we have an RM $\iota \circ P:{\mathbb{S}}^{15}\to {\mathbb{S}}^9$ that satisfies Theorem 2.

Example 3. 

Consider Hopf fibration $\iota :{\mathbb{S}}^7\to {\mathbb{S}}^4$ and the canonical embedding P of a hypersurface ${\mathbb{S}}^4$ of ${\mathbb{R}}^5$ to be $P:{\mathbb{S}}^4\to {\mathbb{R}}^5$. Then, we have a totally umbilical RM $P\circ \iota :{\mathbb{S}}^7\to {\mathbb{R}}^5$ that satisfies Theorem 2 (see [16]).

Corollary 1. 

Let P be a RM from an $(B,g_B)$ onto an s-dimensional Euclidean space $\mathbb{E}$ with $1<rank(P)=p<s$. Then, we have

$$\begin{array}{c}\hfill {\left|\right|trace(h)\left|\right|}^2\ge p^2({\rho }^{\perp }+{\rho }^{nor}).\end{array}$$

By applying Theorem 2 in the specific case in which P acts as an isometric immersion, we are able to retrieve the traditional DDVV inequality for submanifolds in real space forms [3]. Since P is an isometric immersion, we have ${\left|\right|\mathcal{H}\left|\right|}^2=p^2{\left|\right|trace(h)\left|\right|}^2$, and hence, we deduce the following result using Theorem 2:

Corollary 2. 

Let P be an isometric immersion from a $(B,g_B)$ into $(F(c),g_F)$. Then, we have

$$\begin{array}{c}\hfill {\left|\right|\mathcal{H}\left|\right|}^2\ge {\rho }^{\perp }+{\rho }^{nor}-c.\end{array}$$

4. Riemannian Maps with a Riemannian Manifold of (Nearly) Quasi-Constant Curvature

B.-Y. Chen and K. Yano say that a Rie. manifold $(F,g_F)$ is of quasi-constant curvature if the following equation holds [17]:

$$\begin{array}{ccc}\hfill g_F(R^F(\hat{X},\hat{Y})\hat{Z},\hat{W})& =& \alpha \{g_F(\hat{X},\hat{W})g_F(\hat{Y},\hat{Z})-g_F(\hat{X},\hat{Z})g_F(\hat{Y},\hat{W})\}\hfill \\ & & +\beta \{T(\hat{Y})T(\hat{W})g_F(\hat{X},\hat{Z})-g_F(\hat{X},\hat{W})T(\hat{Y})\mathcal{P}\hat{Z}\hfill \\ & & +g_F(\hat{Y},\hat{W})T(\hat{X})\mathcal{P}\hat{Z}-T(\hat{X})T(\hat{W})g_F(\hat{Y},\hat{Z})\},\hfill \end{array}$$

(21)

where $\alpha$ and $\beta$ are scalar functions on F, $\mathcal{P}$ is a unit vector field (called a generator), and T is a one-form dual to $\mathcal{P}$, and it is denoted as $F(\alpha ,\beta ,\mathcal{P})$.

Remark 3. 

It becomes evident that the manifold reduces to a real space form of constant curvature α when it is observed that $\beta =0$.

Subsequently, U.C. De and A.K. Gazi expanded the concept from a Rie. manifold of quasi-constant curvature to that of a Rie. manifold of nearly quasi-constant curvature, where the curvature tensor satisfies [18,19] the following:

$$\begin{array}{ccc}\hfill g_F(R^F(\hat{X},\hat{Y})\hat{Z},\hat{W})& =& \alpha \{g_F(\hat{X},\hat{W})g_F(\hat{Y},\hat{Z})-g_F(\hat{X},\hat{Z})g_F(\hat{Y},\hat{W})\}\hfill \\ & & +\beta \{A(\hat{Y},\hat{W})g_F(\hat{X},\hat{Z})-g_F(\hat{X},\hat{W})A(\hat{Y},\hat{Z})\hfill \\ & & +g_F(\hat{Y},\hat{W})A(\hat{X},\hat{Z})-A(\hat{X},\hat{W})g_F(\hat{Y},\hat{Z})\},\hfill \end{array}$$

(22)

where A represents a non-zero symmetric tensor field of type $(0,2)$. We denote it as $F(\alpha ,\beta ,A)$.

Remark 4. 

When $\beta =0$, the manifold becomes a real space form. Furthermore, if $A=T\otimes T$, a Rie. manifold of nearly quasi-constant curvature transitions into a Rie. manifold of quasi-constant curvature.

Theorem 3. 

Let P be an RM from a $(B,g_B)$ onto an s-dimensional Rie. manifold of nearly quasi-constant curvature $(F(\alpha ,\beta ,A,g_F)$ with $1<rank(P)=p<s$. Then, we have

$$\begin{array}{c}\hfill {\left|\right|trace(h)\left|\right|}^2\ge p^2({\rho }^{\perp }+{\rho }^{nor}-\alpha -2{\displaystyle \frac{\beta }{p}}trace(A{|}_P)),\end{array}$$

where$trace(A{|}_P)$signifies the trace of A restricted to P.

Proof. 

We adopt the same steps followed in proving Theorem 2. Here, the main ingredient is obtained by plunging (22) into (10) to find the relation for scalar curvature:

$$\begin{array}{c}\hfill scal={\displaystyle \frac{p(p-1)}{2}}\alpha +\beta (p-1)trace{(A|}_P+\sum _{i<j}(h_{ii}^a{h}_{jj}^a+{(h_{ij}^a)}^2).\end{array}$$

□

Remark 5. 

Indeed, if P is a isometric immersion, then Theorem 3 simplifies to the inequality derived in [20].

Theorem 4. 

Let P be a RM from $(B,g_B)$ onto an s-dimensional Rie. manifold of quasi-constant curvature $(F(\alpha ,\beta ,\mathcal{P}),g_F)$ with $1<rank(P)=p<s$. Then, we have

$$\begin{array}{c}\hfill {\left|\right|trace(h)\left|\right|}^2\ge p^2({\rho }^{\perp }+{\rho }^{nor}-\alpha -2{\displaystyle \frac{\beta }{p}}{\left|\right|\mathcal{P}\left|\right|}^2).\end{array}$$

Proof. 

The proof of this Theorem is obvious, as one can use $A=T\otimes T$ in Theorem 3 or directly by (21) to obtain the desired inequality. □

5. Riemannian Maps to Complex Space Forms

Definition 2. 

([14,21]). Let $(B,g_B)$ be a Riemannian manifold and $(F,J,g_F)$ be an almost Hermitian manifold. Then, a Riemannian map $P:(B,g_B)\to (F,J,g_F)$ is said to be an anti-invariant Riemannian map at $x\in B$ if $J(range(P_{*x}))\subset {(range(P_{*x}))}^{\perp }$. Note that if P is anti-invariant for every $x\in B$, then P is said to be an anti-invariant Riemannian map. It is convincing to say that an anti-invariant Riemannian map is Lagrangian if $rank(P)=p=dim({(range(P_{*}))}^{\perp })$.

Theorem 5. 

Let P be a Lagrangian RM from an r-dimensional Rie. manifold $(B,g_B)$ onto a complex space form $(F(c),J,g_F)$ of constant holomorphic sectional curvature c with $dim(F)=s>1$ and $1<rank(P)=p$. Then, we have

$$\begin{array}{c}\hfill {({\rho }^{\perp })}^2\le {\displaystyle \frac{2c^2}{p(p-1)}}+({\displaystyle \frac{1}{p^2}}\left|\right|trace(h){\left|\right|}^2-{\rho }^{nor}{+c)}^2+{\displaystyle \frac{4c({\rho }^{nor}-c)}{p(p-1)}}.\end{array}$$

Proof. 

P is a Lagrangian RM from $(B,g_B)$ to a complex space form and its Riemannian curvature tensor is given by

$$\begin{array}{ccc}\hfill g_F(R^F(\hat{X},\hat{Y})\hat{Z},\hat{W})& =& {\displaystyle \frac{c}{4}}\{g_F(\hat{X},\hat{W})g_F(\hat{Y},\hat{Z})-g_F(\hat{X},\hat{Z})g_F(\hat{Y},\hat{W})\hfill \\ & & +g_F(\hat{X},J\hat{Z})g_F(J\hat{Y},\hat{W})-g_F(\hat{Y},J\hat{Z})g_F(J\hat{X},\hat{W})\hfill \\ & & +2g_F(\hat{X},J\hat{Y})g_F(J\hat{Z},\hat{W})\},\hfill \end{array}$$

(23)

for the vector fields $\hat{X},\hat{Y},\hat{Z},\hat{W}$ on F. Next, we choose $\left\{v_i\right|i=1,2,3,\cdots ,p\}$ as an orthonormal base of $H_x$ and $\{V_i=J{P}_{*}{v}_i|i=1,2,3,\cdots ,p\}$ as that of ${(range(P_{*x}))}^{\perp }$. Note that $p=s$. Then, we use the Ricci equation and (23) give the following:

$$\begin{array}{ccc}\hfill sca{l}^{\perp }& =& \sum _{1\le a<b\le p}\sum _{1\le i<j\le p}{g}_F(R^{F\perp }(v_i,v_j)V_a,V_b)\hfill \\ & =& \sum _{1\le a<b\le p}\sum _{1\le i<j\le p}[-c({\delta }_{ia}{\delta }_{jb}-{\delta }_{ib}{\delta }_{ja})+g_F([S_{V_a},S_{V_b}]v_i,v_j)].\hfill \end{array}$$

Also, we have

$$\begin{array}{ccc}\hfill {(sca{l}^{\perp })}^2& =& \sum _{1\le a<b\le p}\sum _{1\le i<j\le p}{g}_F^2(R^{F\perp }(v_i,v_j)V_a,V_b)\hfill \\ & =& \sum _{1\le a<b\le p}\sum _{1\le i<j\le p}{\left[-c({\delta }_{ia}{\delta }_{jb}-{\delta }_{ib}{\delta }_{ja})+g_F([S_{V_a},S_{V_b}]v_i,v_j)\right]}^2\hfill \\ & =& {\displaystyle \frac{p(p-1)c^2}{2}}+\sum _{1\le a<b\le p}\sum _{1\le i<j\le p}{g}_F^2([S_{V_a},S_{V_b}]v_i,v_j)\hfill \\ & & -2c\sum _{1\le a<b\le p}\sum _{1\le i<j\le p}({\delta }_{ia}{\delta }_{jb}-{\delta }_{ib}{\delta }_{ja})g_F([S_{V_a},S_{V_b}]v_i,v_j).\hfill \end{array}$$

(24)

Here, we use the normalized scalar normal curvature, denoted by $sca{l}^{nor}$, as

$$\begin{array}{c}\hfill sca{l}^{nor}={\displaystyle \frac{2}{p(p-1)}}{\left[\sum _{1\le a<b\le p}\sum _{1\le i<j\le p}{g}_F^2([S_{V_a},S_{V_b}]v_i,v_j)\right]}^{1/2}.\end{array}$$

Thus, (24) becomes

$$\begin{array}{ccc}\hfill {(sca{l}^{\perp })}^2& =& {\displaystyle \frac{p(p-1)c^2}{2}}+{\displaystyle \frac{p^2{(p-1)}^2}{4}}{(sca{l}^{nor})}^2\hfill \\ & & -2c\sum _{1\le a<b\le p}\sum _{1\le i<j\le p}({\delta }_{ia}{\delta }_{jb}-{\delta }_{ib}{\delta }_{ja})g_F([S_{V_a},S_{V_b}]v_i,v_j)\hfill \\ & =& {\displaystyle \frac{p(p-1)c^2}{2}}+{\displaystyle \frac{p^2{(p-1)}^2}{4}}{(sca{l}^{nor})}^2-{c\left|\right|h\left|\right|}^2+c\left|\right|trace(h){\left|\right|}^2\hfill \\ & =& {\displaystyle \frac{p(p-1)c^2}{2}}+{\displaystyle \frac{p^2{(p-1)}^2}{4}}{(sca{l}^{nor})}^2+cp(p-1)({\rho }^{nor}-c),\hfill \end{array}$$

which can be rewritten as

$$\begin{array}{c}\hfill {({\rho }^{\perp })}^2={\displaystyle \frac{2c^2}{p(p-1)}}+{(sca{l}^{nor})}^2+{\displaystyle \frac{4c({\rho }^{nor}-c)}{p(p-1)}}.\end{array}$$

(25)

To introduce inequality into (25), we use an inequality obtained in [5] as follows:

$$\begin{array}{c}\hfill \sum _{a=1}^p\sum _{1\le i<j\le p}{(h_{ii}^a-h_{jj}^a)}^2+2p\sum _{a=1}^p\sum _{1\le i<j\le p}(h_{ij}^a)\ge 2p{\left[\sum _{1\le a<b\le p}\sum _{1\le i<j\le p}{\left(\sum _{k=1}^p(h_{jk}^a{h}_{ik}^b-h_{ik}^a{h}_{jk}^b)\right)}^2\right]}^{1/2}.\end{array}$$

By using this inequality, as well as (23) and (10), we derive

$$\begin{array}{ccc}\hfill {\left|\right|trace(h)\left|\right|}^2& \ge & p^{2}sca{l}^{nor}+\sum _{a=1}^p\sum _{1\le i<j\le p}[h_{ii}^a{h}_{jj}^a-{(h_{ij}^a)}^2]\hfill \\ & =& p^{2}sca{l}^{nor}+{\displaystyle \frac{2p}{p-1}}scal-p^{2}c\hfill \\ & =& p^2(sca{l}^{nor}-c)+{\rho }^{nor}.\hfill \end{array}$$

Further, we have

$$\begin{array}{c}\hfill sca{l}^{nor}\le {\displaystyle \frac{1}{p^2}}{\left|\right|trace(h)\left|\right|}^2-{\rho }^{nor}+c.\end{array}$$

(26)

On combining (25) and (26), we conclude our desired relation. □

Corollary 3. 

Let P be a Lagrangian RM from a r-dimensional Rie. manifold $(B,g_B)$ onto a complex Euclidean space $\mathbb{C}$ with $dim(\mathbb{C})=s>1$ and $1<rank(P)=p$. Then, we have

$$\begin{array}{c}\hfill {({\rho }^{\perp })}^2\le ({\displaystyle \frac{1}{p^2}}\left|\right|trace(h){\left|\right|}^2-{\rho }^{nor}{)}^2.\end{array}$$

Applying Theorem 5 specifically to the case where the Lagrangian Riemannian map P is an isometric Lagrangian immersion, we derive the following principal result from [7], called the generalized Wintgen inequality for Lagrangian submanifolds in complex space forms.

Corollary 4. 

Let P be an isometric Lagrangian immersion of $(B,g_B)$ into $(F(c),J,g_F)$. Then, we have

$$\begin{array}{c}\hfill {({\rho }^{\perp })}^2\le {\displaystyle \frac{2c^2}{p(p-1)}}+{(|\left|\mathcal{H}\right||}^2-{\rho }^{nor}{+c)}^2+{\displaystyle \frac{4c({\rho }^{nor}-c)}{p(p-1)}}.\end{array}$$

Example 4. 

Recall an example (from [16]) of an anti-invariant Riemannian map ${\pi }_1\circ {\pi }_2:{\mathbb{RP}}^n(c/4){\times }_{f}F\to {\mathbb{CP}}^n(c)$ between the warped product (of the real projective space ${\mathbb{RP}}^n(c/4)$ of dimension $n>2$ and an arbitrary Rie. manifold F) and the complex projective space of dimension n that satisfies Theorem 5, $f>0$. Here, the mapping ${\pi }_1:{\mathbb{RP}}^n(c/4)\to {\mathbb{CP}}^n(c)$ is canonically imbedded as a totally real and totally geodesic submanifold, and the canonical projection ${\pi }_2:{\mathbb{RP}}^n(c/4){\times }_{f}F\to {\mathbb{RP}}^n(c/4)$ is defined by ${\pi }_1(x,y)=x$ for all $(x,y)\in {\mathbb{RP}}^n(c/4)\times F$ where $f>0$.

6. Concluding Remarks

Riemannian maps are used to study the relationships between different geometric structures on manifolds. They help us with understanding how geometric properties (like curvature) are transferred from one manifold to another and have applications in theoretical physics, particularly in general relativity and the study of space-time manifolds.

The field of Riemannian maps represents a recent area of research, extending beyond traditional studies of Riemannian submanifolds and Riemannian submersions. While much attention has been given to curvature relationships concerning submanifolds and Riemannian submersions, this paper aims to explore Riemannian maps through the lens of curvature relations. The examples that we have presented in this article illustrate the efficacy of our results. By demonstrating the consistency of our findings in specific examples, we offer compelling evidence for the broad applicability and resilience of our results. Although we have established Wintgen inequality for Riemannian maps, there are still many research problems to investigate for interested readers. In this direction, one can extend the derived inequality in the case of Riemannian maps to known space forms, for example, Sasakian space forms and quaternionic space forms (see [22,23]). Furthermore, we want to intersect other techniques and theories to obtain more new results. For example, the new works relevant to singularity theory [24], soliton theory [25,26], hypersurfaces problems [27,28], Chen–Ricci inequality [29,30], submanifolds theory [31,32], tangent bundle problems [33,34], etc. are useful for future research. We will combine the techniques and results mentioned above to explore new approaches for obtaining new results in future research.