Abstract
The Wintgen inequality (1979) is a sharp geometric inequality for surfaces in the 4-dimensional Euclidean space involving the Gauss curvature (intrinsic invariant) and the normal curvature and squared mean curvature (extrinsic invariants), respectively. De Smet et al. (Arch. Math. (Brno) 35:115–128, 1999) conjectured a generalized Wintgen inequality for submanifolds of arbitrary dimension and codimension in Riemannian space forms. This conjecture was proved by Lu (J. Funct. Anal. 261:1284–1308, 2011) and by Ge and Tang (Pac. J. Math. 237:87–95, 2008), independently. In the present paper we establish a generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature.
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1 Introduction
For surfaces \(M^2\) of the Euclidean space \(\mathbb {E}^3\), the Euler inequality \(G\le \Vert H\Vert ^2\) is fulfilled, where G is the (intrinsic) Gauss curvature of \(M^2\) and \(\Vert H\Vert ^2\) is the (extrinsic) squared mean curvature of \(M^2\).
Furthermore, \(G=\Vert H\Vert ^2\) everywhere on \(M^2\) if and only if \(M^2\) is totally umbilical, or still, by a theorem of Meusnier, if and only if \(M^2\) is (a part of) a plane \(\mathbb {E}^2\) or, it is (a part of) a round sphere \( S^2\) in \(\mathbb {E}^3\).
In 1979, Wintgen [25] proved that the Gauss curvature G, the squared mean curvature \(\left\| H\right\| ^2\) and the normal curvature \(G^{\bot }\) of any surface \(M^{2}\) in \(\mathbb {E}^{4}\) always satisfy the inequality
the equality holds if and only if the ellipse of curvature of \(M^{2}\) in \(\mathbb {E}^{4}\) is a circle.
The Whitney 2-sphere satisfies the equality case of the Wintgen inequality identically.
A survey containing recent results on surfaces satisfying identically the equality case of Wintgen inequality can be read in [5].
Later, the Wintgen inequality was extended by Rouxel [20] and by Guadalupe and Rodriguez [10] independently, for surfaces \(M^{2}\) of arbitrary codimension m in real space forms \(\widetilde{M}^{2+m}(c)\); namely
The equality case was also investigated.
A corresponding inequality for totally real surfaces in n-dimensional complex space forms was obtained in [13]. The equality case was studied and a non-trivial example of a totally real surface satisfying the equality case identically was given.
In 1999, De Smet et al. [7] formulated the conjecture on Wintgen inequality for submanifolds of real space forms, which is also known as the DDVV conjecture.
This conjecture was proven by the authors for submanifolds \(M^n\) of arbitrary dimension \(n\ge 2\) and codimension 2 in real space forms \(\tilde{M}^{n+2}(c)\) of constant sectional curvature c.
Recently, the DDVV conjecture was finally settled for the general case by Lu [12] and independently by Ge and Tang [9].
One of the present authors obtained generalized Wintgen inequalities for Lagrangian submanifolds in complex space forms [14] and Legendrian submanifolds in Sasakian space forms [15], respectively. Moreover, two of the present authors established in [3] a version of the Euler inequality and the Wintgen inequality for statistical surfaces in statistical manifolds of constant curvature.
In this paper, using the sectional curvature defined in [19], we derive a generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature.
2 Statistical manifolds and their submanifolds
A statistical manifold is a Riemannian manifold \((\tilde{M}^{n+k},\tilde{g})\) of dimension \((n+k),\) endowed with a pair of torsion-free affine connections \(\tilde{\nabla }\) and \(\tilde{\nabla }^{*}\) satisfying
for any \(X,Y,Z\in \Gamma (T\tilde{M}).\) The connections \(\tilde{\nabla }\) and \(\tilde{\nabla }^{*}\) are called dual connections (see [1, 17, 22]), and it is easily shown that \((\tilde{\nabla }^{*})^{*}=\tilde{\nabla }.\) The pair \((\tilde{\nabla },\tilde{g})\) is said to be a statistical structure. If \((\tilde{\nabla },\tilde{g})\) is a statistical structure on \(\tilde{M}^{n+k},\) so is \((\tilde{\nabla }^{*},\tilde{g})\) [1, 24].
On the other hand, any torsion-free affine connection \(\tilde{\nabla }\) always has a dual connection given by
where \(\tilde{\nabla }^{0}\) is Levi-Civita connection on \(\tilde{M}^{n+k}\).
Denote by \(\tilde{R}\) and \(\tilde{R}^{*}\) the curvature tensor fields of \(\tilde{\nabla }\) and \(\tilde{\nabla }^{*},\) respectively.
A statistical structure \((\tilde{\nabla },\tilde{g})\) is said to be of constant curvature \(c\in \mathbb {R}\) if
A statistical structure \((\tilde{\nabla },\tilde{g})\) of constant curvature 0 is called a Hessian structure.
The curvature tensor fields \(\tilde{R}\) and \(\tilde{R}^{*}\) of dual connections satisfy
From (2.4) it follows immediately that if \((\tilde{\nabla },\tilde{g})\) is a statistical structure of constant curvature c, then \((\tilde{\nabla }^{*},\tilde{g})\) is also a statistical structure of constant curvature c. In particular, if \((\tilde{\nabla },\tilde{g})\) is Hessian, so is \((\tilde{\nabla }^{*},\tilde{g})\) [8].
On a Hessian manifold (\(\tilde{M}^{n+k},\tilde{\nabla })\), let \(\gamma =\tilde{\nabla }^0-\tilde{\nabla }\). The tensor field Q of type (1,3) defined by the covariant differential \(Q=\tilde{\nabla }\gamma \) of \(\gamma \) is said to be the Hessian curvature tensor for \(\tilde{\nabla }\) (see [21]).
By using the Hessian curvature tensor Q, a Hessian sectional curvature can be defined on a Hessian manifold.
A Hessian manifold has constant Hessian sectional curvature \(\tilde{c}\) if and only if (see [21])
for all vector fields on \(\tilde{M}^{n+k}\).
If \((\tilde{M}^{n+k},\tilde{g})\) is a statistical manifold and \(M^n\) a submanifold of dimension n of \(\tilde{M}^{n+k},\) then \((M^{n},g)\) is also a statistical manifold with the induced connection by \(\tilde{\nabla }\) and induced metric g. In the case that \((\tilde{M}^{n+k},\tilde{g})\) is a semi-Riemannian manifold, the induced metric g has to be non-degenerate. For details, see [23, 24].
In the geometry of Riemannian submanifolds (see [4]), the fundamental equations are the Gauss and Weingarten formulas and the equations of Gauss, Codazzi and Ricci.
Let denote the set of the sections of the normal bundle to \(M^{n}\) by \(\Gamma (TM^{n\perp }).\)
In our case, for any \(X,Y\in \Gamma (TM^{n}),\) according to [24], the corresponding Gauss formulas are
where h, \(h^{*}:\Gamma (TM^n)\times \Gamma (TM^n)\rightarrow \Gamma ({TM^n}^\perp )\) are symmetric and bilinear, called the imbedding curvature tensor of \(M^{n}\) in \(\tilde{M}^{n+k}\) for \(\tilde{\nabla }\) and the imbedding curvature tensor of \(M^{n}\) in \(\tilde{M}^{n+k}\) for \(\tilde{\nabla }^{*}, \) respectively.
In [24], it is also proved that \((\nabla ,g)\) and \((\nabla ^{*},g)\) are dual statistical structures on \(M^{n}.\)
Since h and \(h^{*}\) are bilinear, we have the linear transformations \( A_{\xi }\) and \(A_{\xi }^{*}\) on \(TM^n\) defined by
for any \(\xi \in \Gamma (TM^{n\perp })\) and \(X,Y\in \Gamma (TM^{n}).\) Further, see [24], the corresponding Weingarten formulas are
for any \(\xi \in \Gamma (TM^{n\perp })\) and \(X\in \Gamma (TM^{n}).\) The connections \(\nabla _{X}^{\perp }\) and \(\nabla _{X}^{*\perp }\) given by (2.9) and (2.10) are Riemannian dual connections with respect to induced metric on \(\Gamma (TM^{n\perp }).\)
Let \(\{e_{1},\ldots ,e_{n}\}\) and \(\{\xi _{1},\ldots ,\xi _{k}\}\) be orthonormal tangent and normal frames, respectively, on \(M^{n}\). Then the mean curvature vector fields are defined by
and
for \(1\le i,j\le n\) and \(1\le \alpha \le k\) (see also [6]).
The corresponding Gauss, Codazzi and Ricci equations are given by the following result.
Proposition 2.1
[24] Let \(\tilde{\nabla }\) and \(\tilde{\nabla }^*\) be dual connections on \(\tilde{M}^{n+k}\) and \(\nabla \) the induced connection by \(\tilde{\nabla }\) on \(M^{n}.\) Let \(\tilde{R}\) and R be the Riemannian curvature tensors for \(\tilde{\nabla }\) and \(\nabla ,\) respectively. Then,
where \(R^{\perp }\) is the Riemannian curvature tensor of \(\nabla ^\perp \) on \( TM^{n\perp },\xi ,\eta \in \Gamma ( TM^{n\perp })\) and \([A_{\xi }^{*},A_{\eta }]=A_{\xi }^{*}A_{\eta }-A_{\eta }A_{\xi }^{*}.\)
For the equations of Gauss, Codazzi and Ricci with respect to the connection \(\tilde{\nabla }^{*}\) on \(M^{n}\), we have
Proposition 2.2
[24] Let \(\tilde{\nabla }\) and \(\tilde{\nabla }^*\) be dual connections on \(\tilde{M}^{n+k}\) and \(\nabla ^{*}\) the induced connection by \(\tilde{\nabla }^*\) on \(M^{n}.\) Let \(\tilde{R}^{*}\) and \(R^{*}\) be the Riemannian curvature tensors for \(\tilde{\nabla }^{*}\) and \(\nabla ^{*},\) respectively. Then,
where \(R^{*\perp }\) is the Riemannian curvature tensor of \(\nabla ^{\perp *}\) on \(TM^{n\perp }, \xi ,\eta \in \Gamma ( TM^{n\perp })\) and \([A_{\xi },A_{\eta }^{*}]=A_{\xi }A_{\eta }^{*}-A_{\eta }^{*}A_{\xi }.\)
Geometric inequalities for statistical submanifolds in statistical manifolds with constant curvature were obtained in [2].
3 Statistical surfaces in statistical manifolds of constant curvature
Let \((\tilde{M}^{3},\tilde{g})\) be a 3-dimensional statistical manifold of constant curvature c and \(M^{2}\) a surface of \(\tilde{M}.\) Denote the Gauss curvature, the mean curvature and the dual mean curvature of M, by G, H and \(H^{*},\) respectively. In [3], a version of the Euler inequality for statistical surfaces was given.
Proposition 3.1
[3] Let \(M^{2}\) be a surface in a 3-dimensional statistical manifold of constant curvature c. Then its Gauss curvature satisfies :
Some examples of statistical surfaces satisfying the equality case of the above Euler inequality can be provided by the following.
Example 1
(A trivial example) Recall Lemma 5.3 of Furuhata [8].
Let \((\mathbb {H},\tilde{\nabla },\tilde{g})\) be a Hessian manifold of constant Hessian sectional curvature \(\tilde{c}\ne 0,\) \(( M,\nabla ,g)\) a trivial Hessian manifold and \(f:M\longrightarrow \mathbb {H}\) a statistical immersion of codimension one. Then one has:
Thus, if dim \(M=2\), the immersion f of codimension one satisfies the equality case of the statistical version of the Euler inequality given by Proposition 3.1.
Example 2
Let \((\mathbb {H}^{3},\tilde{g})\) be the upper half space of constant sectional curvature \(-1\), i.e.,
An affine connection \(\tilde{\nabla }\) on \(\mathbb {H}^3\) is given by
where \(i,j=1,2.\) The curvature tensor field \(\tilde{R}\) of \(\tilde{\nabla }\) is identically zero, i.e., \(c=0\). Thus \(( \mathbb {H}^{3},\tilde{\nabla }, \tilde{g})\) is a Hessian manifold of constant Hessian sectional curvature 4 (see [21]).
Now let consider a horosphere \(M^{2}\) in \(\mathbb {H}^{3}\) having null Gauss curvature, i.e., \(G\equiv 0\) (for details, see [11]). If \( f:M^{2}\longrightarrow \mathbb {H}^{3}\) is a statistical immersion of codimension one, then, by using Lemma 4.1 of [16], we deduce \(A^{*}=0\), and then \(H^{*}=0.\) This implies that the horosphere \(M^{2}\) satisfies the equality case of the statistical version of the Euler inequality given by Proposition 3.1.
More generally, let consider a 4-dimensional statistical manifold of constant curvature c, i.e. \((\tilde{M}^{4},c),\) and a surface \(M^{2}\) of \(\tilde{M}^{4}.\) We respectively denote the Gauss curvature, the normal curvature and the Gauss curvature with respect to the Levi-Civita connection by G, \(G^{\perp }\) and \(G^{0}.\) Similarly, we respectively denote the mean vector field, the dual mean curvature and the sectional curvature with respect to the Levi-Civita connection by H, \(H^{*}\) and \(\tilde{K}^{0}.\) We have the following Wintgen inequalities.
Theorem 3.2
[3] Let \(M^{2}\) be a statistical surface in a 4-dimensional statistical manifold \((\tilde{M}^{4},c)\) of constant curvature c. Then
In particular, for \(c=0\) we derive the following.
Corollary 3.3
[3] Let \(M^{2}\) be a statistical surface of a Hessian 4-dimensional statistical manifold \(\tilde{M}^{4}\) of Hessian curvature 0. Then :
4 Wintgen inequality for statistical submanifolds
Let \(M^{n}\) be an n-dimensional statistical submanifold of a \((n+m)\)-dimensional statistical manifold \((\tilde{M}^{n+m},c) \) of constant curvature c.
The sectional curvature K on \(M^{n}\) is defined by [3] (see also [18, 19])
for any orthonormal vectors \(X,Y\in T_{p}M^{n},\) \(p\in M^{n}.\)
In the case of the Levi-Civita connection, the above definition coincides (up to the sign) to the standard definition of the sectional curvature.
Let \(p\in M^{n}\) and \(\{e_{1},e_{2},\ldots ,e_{n}\}\) an orthonormal basis of \(T_{p}M^{n}.\) Then the normalized scalar curvature \(\rho \) is defined by (see [7]):
By using the Gauss equations for the dual connections \(\tilde{\nabla }\) and \( \tilde{\nabla }^{*}\), respectively, we obtain
Denoting as usual by
the above equation becomes
On the other hand, the normalized normal scalar curvature \(\rho ^{\perp }\) is defined by (see also [3]):
The Ricci equations for the dual connections \(\tilde{\nabla },\) and \(\tilde{ \nabla }^{*}\), respectively, imply
or equivalently,
It follows that
It is known that the components of the second fundamental form \(h^0\) of \(M^n\) with respect to the Levi-Civita connection \(\tilde{\nabla }^0\) are given by \(2h_{ik}^{0r}=h_{ik}^{r}+h_{ik}^{*r},\) \(\forall i,k=1,\ldots ,n,\) \(r=1,\ldots ,m.\) Then we can write
We shall use the algebraic inequality
Therefore
Recall an inequality from [12] (see also [14])
Similarly, we have
and
Summing up the above three inequalities, from (4.3) we obtain
Also, we can write
and similarly,
and
Substituting in (4.4), we get
Using again \(2h_{ij}^{0r}=h_{ij}^{r}+h_{ij}^{*r},\) \(\forall i,j=1,\ldots ,n,\) \(r=1,\ldots ,m,\) we obtain
Substituting (4.1) in (4.5), one leads to
If we denote by
the Gauss equation for the Levi-Civita connection \(\tilde{\nabla }^{0}\) gives
From (4.6) and (4.7) we obtain
Summarizing, we proved the following generalized Wintgen inequality.
Theorem 4.1
Let \(M^{n}\) be a submanifold in a statistical manifold \((\tilde{M}^{n+m},c)\) of constant curvature c. Then
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Acknowledgments
A part of this paper was written during the visit of I. Mihai to Firat University, Turkey, in April to May 2015. He was supported by the Scientific and Technical Research Council of Turkey (TUBITAK) for Advanced Fellowships Programme.
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Aydin, M.E., Mihai, A. & Mihai, I. Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Bull. Math. Sci. 7, 155–166 (2017). https://doi.org/10.1007/s13373-016-0086-1
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DOI: https://doi.org/10.1007/s13373-016-0086-1