Super warped products with a semi-symmetric non-metric connection

1.   Introduction

The (singly) warped product $B\times_hF$ of two pseudo-Riemannian manifolds $(B, g_B)$ and $(F, g_F)$ with a smooth function $h:B\rightarrow (0, \infty)$ is the product manifold $B\times F$ with the metric tensor $g = g_B\oplus h^2g_F.$ Here, $(B, g_B)$ is called the base manifold, $(F, g_F)$ is called as the fiber manifold and $h$ is called as the warping function. Generalized Robertson-Walker space-times and standard static space-times are two well-known warped product spaces. The concept of warped products was first introduced by Bishop and ONeil (see ^[4]) to construct examples of Riemannian manifolds with negative curvature. In Riemannian geometry, warped product manifolds and their generic forms have been used to construct new examples with interesting curvature properties since then. In ^[7], F. Dobarro and E. Dozo had studied from the viewpoint of partial differential equations and variational methods, the problem of showing when a Riemannian metric of constant scalar curvature can be produced on a product manifolds by a warped product construction. In ^[8], Ehrlich, Jung and Kim got explicit solutions to warping function to have a constant scalar curvature for generalized Robertson-Walker space-times. In ^[3], explicit solutions were also obtained for the warping function to make the space-time as Einstein when the fiber is also Einstein.

N. S. Agashe and M. R. Chafle introduced the notion of a semi-symmetric non-metric connection and studied some of its properties and submanifolds of a Riemannian manifold with a semi-symmetric non-metric connection ^[1,2]. In ^[13,14], Sular and Özgur studied warped product manifolds with a semi-symmetric metric connection and a semi-symmetric non-metric connection, they computed curvature of semi-symmetric metric connection and semi-symmetric non-metric connection and considered Einstein warped product manifolds with a semi-symmetric metric connection and a semi-symmetric non-metric connection. In ^[16], Wang studied the Einstein multiply warped products with a semi-symmetric metric connection and the multiply warped products with a semi-symmetric metric connection with constant scalar curvature.

On the other hand, in ^[5], the definition of super warped product spaces was given. Einstein warped products were studied in ^[9]. In ^[10], several new super warped product spaces were given and the authors also studied the Einstein equations with cosmological constant in these new super warped product spaces. In ^[17], Wang studied super warped product spaces with a semi-symmetric metric connection. Our motivation is to study super warped product spaces with a semi-symmetric non-metric connection.

In Section 2, we state some definitions of super manifolds and super Riemannian metrics. We also define a semi-symmetric non-metric connection on super Riemannian manifolds and prove that there is a unique semi-symmetric non-metric connection on super Riemannian manifolds which is non-metric and has the semi-symmetric torsion. In Section 3, we compute the curvature tensor and the Ricci tensor of a semi-symmetric non-metric connection on super warped product spaces. In Section 4, we introduce two kinds of super warped product spaces with a semi-symmetric non-metric connection and give the conditions that two super warped product spaces with a semi-symmetric non-metric connection are the Einstein super spaces with a semi-symmetric non-metric connection.

2.   A semi-symmetric non-metric connection on super Riemannian manifolds

In this section, we give some definitions about Riemannian supergeometry.

Definition 2.1. (Definition 1 in ^[5]) A locally $\mathbb{Z}_2$-ringed space is a pair $S: = (|S|, \mathcal{O}_S)$ where $|S|$ is a second-countable Hausdorff space, and a $\mathcal{O}_S$ is a sheaf of $\mathbb{Z}_2$-graded $\mathbb{Z}_2$-commutative associative unital $\mathbb{R}$-algebras, such that the stalks $\mathcal{O}_{S, p}$, $p\in |S|$ are local rings.

In this context, $\mathbb{Z}_2$-commutative means that any two sections $s, t\in \mathcal{O}_S(|U|), \; \; |U|\subset|S|$ open, of homogeneous degree $|s|\in \mathbb{Z}_2$ and $|t|\in \mathbb{Z}_2$ commute up to the sign rule $st = (-1)^{|s||t|}ts$. $\mathbb{Z}_2$-ring space $U^{m|n}: = (U, C^{\infty}_{U^m}\otimes \wedge \mathbb{R}^n)$, is called standard superdomain where $C^{\infty}_{U^m}$ is the sheaf of smooth functions on $U$ and $\wedge\mathbb{R}^n$ is the exterior algebra of $\mathbb{R}^n$. We can employ (natural) coordinates $x^I: = (x^a, \xi^A)$ on any $\mathbb{Z}_2$-domain, where $x^a$ form a coordinate system on $U$ and the $\xi^A$ are formal coordinates.

Definition 2.2. (Notation and preliminary concepts in ^[6]) A supermanifold of dimension $m|n$ is a super ringed space $M = (|M|, \mathcal{O}_M)$ that is locally isomorphic to $\mathbb{R}^{m|n}$ and $|M|$ is a second countable and Hausdorff topological space.

The tangent sheaf $\mathcal{T}M$ of a $\mathbb{Z}_2$-manifold $M$ is defined as the sheaf of derivations of sections of the structure sheaf, i.e., $\mathcal{T}M(|U|) : = {\rm Der}(\mathcal{O}_M(|U|)),$ for arbitrary open set $|U|\subset |M|.$ Naturally, this is a sheaf of locally free $\mathcal{O}_M$-modules. Global sections of the tangent sheaf are referred to as vector fields. We denote the $\mathcal{O}_M (|M|)$-module of vector fields as ${\rm Vect}(M)$. The dual of the tangent sheaf is the cotangent sheaf, which we denote as $\mathcal{T}^*M$. This is also a sheaf of locally free $\mathcal{O}_M$-modules. Global section of the cotangent sheaf we will refer to as one-forms and we denote the $\mathcal{O}_M(|M|)$-module of one-forms as $\Omega^1(M)$.

Definition 2.3. (Definition 4 in ^[5]) A Riemannian metric on a $\mathbb{Z}_2$-manifold M is a $\mathbb{Z}_2$-homogeneous, $\mathbb{Z}_2$-symmetric, non-degenerate, $\mathcal{O}_M$-linear morphisms of sheaves $\left < -, -\right > _g:\; \; \mathcal{T}M\otimes \mathcal{T}M\rightarrow \mathcal{O}_M.$ A $\mathbb{Z}_2$-manifold equipped with a Riemannian metric is referred to as a Riemannian $\mathbb{Z}_2$-manifold.

We will insist that the Riemannian metric is homogeneous with respect to the $\mathbb{Z}_2$-degree, and we will denote the degree of the metric as $|g| \in \mathbb{Z}_2$. Explicitly, a Riemannian metric has the following properties:

(1) $|\left < X, Y\right > _g | = |X| + |Y |+ |g|,$

(2) $\left < X, Y\right > _g = (-1)^{|X||Y|}\left < Y, X\right > _g,$

(3) If $\left < X, Y\right > _g = 0$ for all $Y \in Vect(M),$ then $X = 0,$

(4) $\left < fX+Y, Z\right > _g = f\left < X, Z\right > _g +\left < Y, Z\right > _g,$

for arbitrary (homogeneous) $X, Y, Z \in {\rm Vect}(M)$ and $f \in C^{\infty}(M)$. We will say that a Riemannian metric is even if and only if it has degree zero. Similarly, we will say that a Riemannian metric is odd if and only if it has degree one. Any Riemannian metric we consider will be either even or odd as we will only be considering homogeneous metrics.

recall the definition of the warped product of Riemannian $\mathbb{Z}_2$-manifolds. For details, see the Section 2.3 in ^[5]. Let $M_1\times M_2$ be the product of two $\mathbb{Z}_2$-manifolds $M_1$ and $M_2$. Let $(M_i, g_i)(i = 1, 2)$ be Riemannian $\mathbb{Z}_2$-manifolds whose Riemannian metric are of the same $\mathbb{Z}_2$-degree. Let $\mu\in C^{\infty} (M_1)$ be a degree $0$ invertible global functions that is strictly positive, i.e., $\varepsilon_{M_1}(\mu)$ a strictly positive function on $|M_1|$ where $\varepsilon$ is simply "throwing away" the formal coordinates. Then the warped product is defined as

where $\pi_i: M_1\times M_2\rightarrow M_i\; \; (i = 1, 2)$ is the projection. By Proposition 4 in ^[5], the warped product $M_1\times_\mu M_2$ is a Riemannian $\mathbb{Z}_2$-manifold.

Definition 2.4. (Definition 9 in ^[5]) An affine connection on a $\mathbb{Z}_2$-manifold is a $\mathbb{Z}_2$-degree preserving map

which satisfies the following

1) Bi-linearity

2) $C^{\infty}(M)$-linearrity in the first argument

3) The Leibniz rule

for all homogeneous $X, Y, Z\in {\rm Vect}(M)$ and $f\in C^{\infty}(M)$.

Definition 2.5. (Definition 10 in ^[5]) The torsion tensor of an affine connection

$T_\nabla:\; \; {\rm Vect}(M)\otimes_{C^{\infty}(M)} {\rm Vect}(M)\rightarrow {\rm Vect}(M)$ is defined as

for any (homogeneous) $X, Y \in {\rm Vect}(M)$. An affine connection is said to be symmetric if the torsion vanishes.

Definition 2.6. (Definition 11 in ^[5]) An affine connection on a Riemannian $\mathbb{Z}_2$-manifold $(M, g)$ is said to be metric compatible if and only if

for any $X, Y, Z\in {\rm Vect}(M)$.

Theorem 2.7. (Theorem 1 in ^[5]) There is a unique symmetric (torsionless) and metric compatibleaffine connection $\nabla^L$ on a Riemannian $\mathbb{Z}_2$-manifold $(M, g)$ which satisfies the Koszul formula

for all homogeneous $X, Y, Z\in {\rm Vect}(M)$.

Definition 2.8. (Definition 13 in ^[5]) The Riemannian curvature tensor of an affine connection

is defined as

for all $X, Y$ and $Z \in {\rm Vect}(M)$.

Directly from the definition it is clear that

for all $X, Y$ and $Z \in {\rm Vect}(M)$.

Definition 2.9. (Definition 14 in ^[5]) The Ricci curvature tensor of an affine connection is the symmetric rank-$2$ covariant tensor defined as

where $X, Y\in {\rm Vect}(M)$ and $[\; \;]^I$ denotes the coefficient of $\partial_{x^I}$ and $\partial_{x^I}$ is the natural frame of $\mathcal{T}M$.

Definition 2.10. (Definition 16 in ^[5]) Let $f \in C^{\infty}(M)$ be an arbitrary function on a Riemannian $\mathbb{Z}_2$-manifold $(M, g)$. The gradient of $f$ is the unique vector field ${\rm grad}_gf$ such that

for all $X \in {\rm Vect}(M)$.

Definition 2.11. (Definition 17 in ^[5]) Let $(M, g)$ be a Riemannian $\mathbb{Z}_2$-manifold and let $\nabla^L$ be the associated Levi-Civita connection. The covariant divergence is the map ${\rm Div}_L:{\rm Vect}(M)\rightarrow C^{\infty}(M)$, given by

for any arbitrary $X \in {\rm Vect}(M)$.

Definition 2.12. (Definition 18 in ^[5]) Let $(M, g)$ be a Riemannian $\mathbb{Z}_2$-manifold and let $\nabla^L$ be the associated Levi-Civita connection. The connection Laplacian (acting on functions) is the differential operator of $\mathbb{Z}_2$-degree $|g|$ defined as

for any and all $f \in C^{\infty}(M).$

Definition 2.13. Let $(M, g)$ be a Riemannian $\mathbb{Z}_2$-manifold and $P\in {\rm Vect}(M)$ which satisfied $|g|+|P| = 0$ and we define a semi-symmetric non-metric connection $\widehat{\nabla}$ on $(M, g)$

for any homogenous $X, Y\in {\rm Vect}(M)$ and where $X\cdot f = (-1)^{|X||f|}fX$ for $f\in C^{\infty}(M)$.

Obviously, we have $\widehat{\nabla}_{X+Y}Z = \widehat{\nabla}_{X}Z+\widehat{\nabla}_{Y}Z; \; \; \widehat{\nabla}_{X}(Y+Z) = \widehat{\nabla}_XY+\widehat{\nabla}_XZ,$ for any homogenous $X, Y, Z\in {\rm Vect}(M)$. We can verify that $\widehat{\nabla}_XY$ satisfies the Definition 2.4, then $\widehat{\nabla}_XY$ is an affine connection. By Definition 2.5, we get

Then, we call that $\widehat{\nabla}_XY$ is a semi-symmetric connection. By Definition 2.6 and Definition 2.13, we get

So $\widehat{\nabla}$ doesn't preserve the metric.

Theorem 2.14. There is a unique non-metric compatibleaffine connection $\widehat{\nabla}$ on a Riemannian $\mathbb{Z}_2$-manifold $(M, g)$ which satisfies (2.9) and (2.10).

Proof. By (2.9), we know that a semi-symmetric non-metric connection $\widehat{\nabla}$ satisfies the conditions in Theorem 2.14, then we only need to prove the uniqueness. Let $\nabla^*$ be the other connection which satisfies (2.9) and (2.10). And let $\nabla^*_XY = \nabla^L_XY+B(X, Y),$ then

By $\nabla^L$ preserving the metric and (2.10), we get

So

By $\nabla^L$ having no torsion, we have

By (2.13) and (2.14) and $|B| = 0$, we have

By (2.9) and (2.15), we get

then $B(X, Y) = X\cdot g(Y, P)$. So $\nabla^* = \widehat{\nabla}$, we get the proof of uniqueness.

Proposition 2.15. The following equality holds

where $\pi$ is a one form defined by $\pi(Z): = g(Z, P)$ and $|\pi| = 0$.

Proof. By Definition 2.8 and Definition 2.13, we have

since $\nabla^L$ preserving metric, we have

Then, by (2.17) and (2.18), we can get Proposition 2.15.

3.   Super warped products with a semi-symmetric non-metric connection

Let $(M = M_1\times_\mu M_2, g_\mu = \pi^*_1 g_1+\pi^*_1(\mu)\pi_2^*g_2)$ be the super warped product with $|g| = |g_1| = |g_2|$ and $|\mu| = 0$. For simplicity, we assume that $\mu = h^2$ with $|h| = 0$. Let $\nabla^{L, \mu}$ be the Levi-Civita connection on $(M, g_\mu)$ and $\nabla^{L, M_1}$ (resp. $\nabla^{L, M_2}$) be the Levi-Civita connection on $(M_1, g_1)$ (resp. $(M_2, g_2)$).

Lemma 3.1. (Lemma 3.1 in ^[17]) For $X, Y, Z\in{\rm Vect}(M_1)$ and $U, W, V\in {\rm Vect}(M_2)$, we have

Let $R^{L, \mu}$ denote the curvature tensor of the Levi-Civita connection on $(M, g_\mu)$ and let $R^{L, M_1}$ (resp. $R^{L, M_2}$) be the curvature tensor of the Levi-Civita connection on $(M_1, g_1)$ (resp. $(M_2, g_2)$). Let $H^h_{M_1}(X, Y): = XY(h)-\nabla^{L, {M_1}}_XY(h)$, then $H^h_{M_1}(fX, Y) = fH^h_{M_1}(X, Y)$ and $H^h_{M_1}(X, fY) = (-1)^{|f||X|}fH^h_{M_1}(X, Y)$. $H^h_{M_1}$ is a $(0, 2)$ tensor.

Proposition 3.2. (Proposition 3.2 in ^[17]) For $X, Y, Z\in{\rm Vect}(M_1)$ and $U, V, W\in {\rm Vect}(M_2)$, we have

For $\overline{X}, \overline{Y}, {P}\in {\rm Vect}(M)$, we define

For ${X}, {Y}, {P}\in {\rm Vect}(M_1)$, we define

By Lemma 3.1, (3.3) and (3.4), we have

Lemma 3.3. For $X, Y, P\in{\rm Vect}(M_1)$ and $U, W\in {\rm Vect}(M_2)$ and $\pi(X) = g_1(X, P)$, we have

Lemma 3.4. For $X, Y\in{\rm Vect}(M_1)$ and $U, W, P\in {\rm Vect}(M_2)$, we have

By Proposition 2.15, Lemmas 3.3 and 3.4, we get the following propositions.

Proposition 3.5. For $X, Y, Z, P\in{\rm Vect}(M_1)$ and $U, V, W\in {\rm Vect}(M_2)$, we have

Proof. (1) By Lemma 3.1 and Definition 2.8, we get

so (1) holds.

(2) By Lemma 3.1 and Proposition 2.15, we have

so we get (2).

(3) By Lemma 3.1 and Proposition 2.15, we have

then we get (3).

(4) Similar to (3), we get $R_{\widehat{\nabla}^\mu}(V, W)X = 0$.

(5) By Lemma 3.1 and Proposition 2.15, we have

and by

so we have

Obviously, we can get

(6) By Lemma 3.1 and Proposition 2.15, we have

by

and

so we have

Obviously, we can get

Proposition 3.6. For $X, Y, Z\in{\rm Vect}(M_1)$ and $U, V, W, P\in {\rm Vect}(M_2)$, we have

In the following, we compute the Ricci tensor of $M$. Let $M_1$ (resp. $M_2$) have the $(p, m)$ (resp. $(q, n)$) dimension. Let $\partial_{x^I} = \{\partial_{x^a}, \partial_{\xi^A}\}$ (resp. $\partial_{y^J} = \{\partial_{y^b}, \partial_{\eta^B}\}$) denote the natural tangent frames on $M_1$ (resp. $M_2$). Let ${\rm Ric}^{L, \mu}$ (resp. ${\rm Ric}^{L, M_1}$, ${\rm Ric}^{L, M_2}$) denote the Ricci tensor of $(M, g_\mu)$ (resp. $(M_1, g_1)$, $(M_2, g_2)$). Then by (2.4), (2.7) and (3.2), we have

Proposition 3.7. The following equalities holds

Let ${\rm Ric}^{\widehat{\nabla}^\mu}$ (resp. ${\rm Ric}^{\widehat{\nabla}^{M_1}}$) denote the Ricci tensor of $(M, \widehat{\nabla}^\mu, g_\mu)$ (resp. $(M_1, \widehat{\nabla}^{M_1}, g_1$). Then by Proposition 3.5, (2.4) and (2.6), we have

Proposition 3.8. The following equalities holds

Proof. (1) By Definition 2.9, we have

so (1) holds.

(2) Similar to (2) in Propsition 3.7, we get

(3) By Definition 2.9, we have

where

by Propsition 3.2, we have

then, we get

so (3) holds.

4.   Special super warped products with a semi-symmetric non-metric connection

In this section, we construct an Einstein super warped product with a semi-symmetric non-metric connection. Let $(M_2^{(q, n)}, g_2)$ be a super Riemannian manifold and $\mathbb{R}^{(1, 0)}$ be the real line. We consider the super Riemannian manifold $M = \mathbb{R}^{(1, 0)}\times _\mu M_2^{(q, n)}$ and $g_\mu = -dt\otimes dt+h^2g_2$, where $h(t)$ and $\mu(t) = h(t)^2$ be non-zero functions for $t\in \mathbb{R}$ and $|g_2| = 0$.

Let $P = \partial_t$, then by Definition 2.8 and Definition 2.9, we get $R_{\widehat{\nabla}^{\mathbb{R}}}(\partial_t, \partial_t)\partial_t = 0$ and ${\rm Ric}^{\widehat{\nabla}^{\mathbb{R}}}(\partial_{t}, \partial_{t}) = 0$. By computations, we have $H^h_{M_1}(\partial_{t}, \partial_{t}) = h''$, ${\rm grad}_{g_1}(h) = -h'\partial_t$ and $\triangle_{g_1}^L(h) = -h''$. By Propsition 3.8, we have

Proposition 4.1. The following equalities holds

Proof. (1) By (1) in Propsition 3.8, we have

(2) By (2) in Propsition 3.8, we have

(3) By (3) in Propsition 3.8, $({\rm grad}_{g_1}h)(h) = -h'\partial_t$ and $\triangle^L_{g_1}(h) = -h''$, we have

Definition 4.2. We call that $(M, g_\mu, \widehat{\nabla}^\mu)$ is Einstein if ${\rm Ric}^{\widehat{\nabla}^\mu}(\overline{X}, \overline{Y}) = \lambda g_\mu(\overline{X}, \overline{Y}),$ for $\overline{X}, \overline{Y}\in {\rm Vect}(M)$ and a constant $\lambda$.

As in the ordinary warped product case (see Theorem 15 in ^[16]), by (4.1) and Definition 4.2, we have the following theorems.

Theorem 4.3. Let $M = \mathbb{R}^{(1, 0)}\times _\mu M_2^{(q, n)}$ and $g_\mu = -dt\otimes dt+h^2g_2$ and $P = \partial_t$. Then $(M, g_\mu, \widehat{\nabla}^\mu)$ is Einstein with the Einstein constant $\lambda$ if and only if the following conditions are satisfied

(1) $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $c_0$.

(2)

(3)

Proof. (1) By (3) in Propsition 4.1, we have

then

by two sides of the Eq (4.8) act simultaneously on $\partial_t$ and $g_2(\partial_{y^L}, \partial_{y^J})\neq 0$, we have $L(t) = c_0$, so ${\rm Ric}^{L, M_2}(\partial_{y^L}, \partial_{y^J}) = c_0g_2(\partial_{y^L}, \partial_{y^J})$, therefore (1) holds.

(2) By (1) in Propsition 4.1 and Definition 4.2, we have

then we get $\lambda = (q-n)(\frac{h''}{h}-1).$

(3) By (1), we get $\lambda h^2-h''h-(q-n-1)(h')^2+(q-n)hh' = c_0$.

By Theorem 4.3, similar to the ordinary warped product case (see Theorem 3.1 in ^[15]), we have

Theorem 4.4. Let $M = \mathbb{R}^{(1, 0)}\times _\mu M_2^{(q, n)}$ and $g_\mu = -dt\otimes dt+h^2g_2$ and $P = \partial_t,$ when $q-n = 1$, then $(M, g_\mu, \nabla^\mu)$ is Einstein with the Einstein constant $-\lambda_0$ if and only if the following conditions are satisfied

(1) $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $c_0 = hh'-h^2$.

(2-1) $\lambda_0 < 1, \; \; f(t) = c_1e^{\sqrt{1-\lambda_0}t}+c_2e^{-\sqrt{1-\lambda_0}t},$

(2-2)$\lambda_0 = 1, \; \; f(t) = c_1+c_2t,$

(2-3) $\lambda_0 > 1, \; \; f(t) = c_1{\cos}\left(\sqrt{\lambda_0-1}t\right)+ c_2{\rm sin}\left(\sqrt{\lambda_0-1}t\right),$

Proof. (1) Let $\lambda = -\lambda_0$ and $c_0 = -\lambda_N$, then

when $q-n = 1$, then

By $(q-n)(\frac{h''}{h}-1) = -\lambda_0,$ we have $\lambda_N = h^2-hh'$ and $h'' = (1-\lambda_0)h$, so $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $c_0 = hh'-h^2$.

(2) By $h'' = (1-\lambda_0)h$, we have characteristic equation $\mu^2-(1-\lambda_0) = 0,$ then $\Delta = 4(1-\lambda_0)$, so (2) holds.

Proposition 4.5. Let $M = \mathbb{R}^{(1, 0)}\times _\mu M_2^{(q, n)}$ and $g_\mu = -dt\otimes dt+h^2g_2$ and $P = \partial_t,$ when $q-n = 0$, then $(M, g_\mu, \widehat{\nabla}^\mu)$ is Einstein with the Einstein constant $-\lambda_0$ if and only if the following conditions are satisfied

(1) $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $c_0$.

(2) $\lambda_0 = 0,$

(3) $c_0+hh''-h'^2 = 0$.

Proof. When $q-n = 0,$ we have $\lambda_0 = 0$ and $\lambda_N-hh''+h'^2 = 0$, then we get Propsition 4.5.

Theorem 4.6. Let $M = \mathbb{R}^{(1, 0)}\times _\mu M_2^{(q, n)}$ and $g_\mu = -dt\otimes dt+h^2g_2$ and $P = \partial_t$, when $q-n\neq 0, 1$, then $(M, g_\mu, \widehat{\nabla}^\mu)$ is Einstein with the Einstein constant $-\lambda_0$ if and only if $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $c_0$ and one of the following conditionsholds

(1) $\lambda_0 = \lambda_N = 0, h = c_1e^t;$

(2) $\lambda_0 = q-n, h = c_1 = \sqrt{\frac{\lambda_N}{q-n}}.$

Proof. Let $\lambda = -\lambda_0$ and $c_0 = -\lambda_N$, then by (4.5), we have $h'' = (1-\frac{\lambda_0}{q-n})h$. By (4.6), we get

when $q-n\neq0, 1,$ we get

Let $q-n = l,$ $\frac{\lambda_0}{q-n} = \frac{\lambda_0}{l} = d_0,$ $\frac{\lambda_N}{1-q+n} = \frac{\lambda_N}{1-L} = \overline{d_0},$

Case (a). When $d_0 < 1,$ let $a_0 = \sqrt{1-d_0},$ $b_0 = -\sqrt{1-d_0},$ then $a_0+b_0 = 0,$ $a_0b_0 = d_0-1$ and $h = c_1e^{a_0t}+c_2e^{b_0t},$ by (4.13), we have

then

Case (a-1). When $c_1 = 0, \; \; c_2\neq 0,$ we get $a_0 = -1, \; \; b_0 = 1,$ then this is a contradiction.

Case (a-2). When $c_1\neq0, \; \; c_2 = 0,$ we get $a_0 = 1, \; \; b_0 = -1, \; \; \lambda_N = 0, \; \; \lambda_0 = 0, \; \; h = c_1e^t.$

Case (a-3). When $c_1\neq0, \; \; c_2\neq0,$ then there is no solution.

Case (b). When $d_0 = 1,$ then $h = c_1+c_2t,$ by (4.13), we have

Case (b-1). When $c_1 = 0, \; \; c_2\neq 0,$ then $\overline{d_0}+c_2^2[1+(d_0-\frac{1}{1-l})t^2+\frac{l}{1-l}t] = 0,$ we get $c_2 = 0,$ this is a contradiction.

Case (b-2). When $c_1\neq0, \; \; c_2 = 0,$ then $\overline{d_0}+c_1^2(d_0-\frac{1}{1-l}) = 0,$ we get $h = c_1 = \frac{\lambda_N}{l}.$

Case (b-3). When $c_1\neq0, \; \; c_2\neq0,$ then $c_2^2(d_0-\frac{1}{1-l}) = 0, \; \; c_2^2\frac{l}{1-l}+2c_1c_2(d_0-\frac{1}{1-l}) = 0,$ we get $c_2 = 0,$ so this is a contradiction.

Case (c). When $d_0 > 1,$ let $h_0 = \sqrt{d_0-1},$ then $h = c_1cosh_0t+c_2sinh_0t,$ by (4.13), we have

then

By (4.18), we can get $c_1 = c_2 = 0,$ so this is a contradiction. Nextly, we give another example. Let $M_1 = \mathbb{R}^{(1, 2)}$ with coordinates $(t, \xi, \eta)$ and $|t| = 0, \; \; |\xi| = |\eta| = 1$. We give a metric $g_1 = -dt\otimes dt+d\xi\otimes d\eta-d\eta\otimes d\xi$ on $M_1$, i.e.,

for the other pair $(\partial_{x^I}, \partial_{x^K})$. Let $\widetilde{M} = \mathbb{R}^{(1, 2)}\times_\mu M_2^{(q, n)}$ and $g_\mu = g_1+h(t)^2g_2$ and $P = \partial_t$. By Proposition 7 in ^[5], we have the Christoffel symbols $\Gamma^L_{JI} = 0$, then

We have

By Proposition 3.7 and the Einstein condition, we have

Theorem 4.7. Let $\widetilde{M} = \mathbb{R}^{(1, 2)}\times _\mu M_2^{(q, n)}$ and $g_\mu = g_1+h^2g_2$ and $P = \partial_t$. Then $(\widetilde{M}, g_\mu, \nabla^{L, \mu})$ is Einstein with the Einstein constant $\lambda$ if and only if one of the following conditions is satisfied

(1) $\lambda = 0$, $q = n$, $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $-c_0$ and $hh''-h'^2 = c_0$.

(2) $\lambda = 0$, $q-n-1 = 0$, $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $0$ and $h = c_1t+c_2$ where $c_1, c_2$ are constant.

(3) $\lambda = 0$, $q-n-1\neq 0, -1$, $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $-c_0$ and$h = \pm \sqrt{\frac{c_0}{q-n-1}}t+c_2$, $\frac{c_0}{q-n-1}\geq 0$.

Proof. By (1) in Propsition 3.7, we have

then

so we get $\lambda = 0$ and $q = n$ or $h'' = 0.$

By $\lambda = 0$, then we have

By (3) in Propsition 3.7 and (4.25), we have

Then we get:

(Case-a). When $\lambda = 0, q = n,$ by $hh''+(q-n-1)h'^2 = c_0,$ we get $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $-c_0$ and $hh''-h'^2 = c_0$.

(Case-b). When $\lambda = 0, h'' = 0,$ by $hh''+(q-n-1)h'^2 = c_0,$ we have $(q-n-1)h'^2 = c_0.$

(Case-b-1). When $q-n-1 = 0,$ $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $0$ and $h = c_1t+c_2$ where $c_1, c_2$ are constant.

(Case-b-2). When $q\neq n,$ $q-n-1\neq0,$ $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $-c_0$ and $h = \pm \sqrt{\frac{c_0}{q-n-1}}t+c_2$, $\frac{c_0}{q-n-1}\geq 0$.

By (2.16) and (4.20), we can get

for other pairs $(\partial_{x^J}, \partial_{x^K}, \partial_{x^L})$. By (2.4) and (4.27), we have

for other pairs $(\partial_{x^J}, \partial_{x^L})$. If $(\widetilde{M}, g_\mu, \widehat{\nabla}^{\mu})$ is Einstein with the Einstein constant $\lambda$, by Propsition 2.15 and (4.28), we have

Solving (4.29), we get

By (3.22) (3) and the Einstein condition, we get $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $c_0$ and

Then we have the following theorem

Theorem 4.8. Let $\widetilde{M} = \mathbb{R}^{(1, 2)}\times _\mu M_2^{(q, n)}$ and $g_\mu = g_1+h^2g_2$ and $P = \partial_t$. Then $(\widetilde{M}, g_\mu, \widehat{\nabla}^{\mu})$ is Einstein with the Einstein constant $\lambda$ if and only if $(M_2^{(q, n)}, \nabla^{L, M_2})$ is Einstein with the Einstein constant $c_0 = 0$ and $\lambda = 0$, $h = c^*$, $q-n+2 = 0$.

Proof. Let $k = 1+\frac{2}{q-n}$, by $\lambda = 0,$ (4.30) and (4.31), we have

Let $b_1 = -(q-n-1)k^2c_1^2+(q-n-2)kc_1^2,$ $b_2 = -(q-n-1)k^2c_2^2+(q-n-2)kc_2^2,$ $b_3 = -c_1k^2,$ $b_4 = -c_2k^2,$ $b_5 = c_0-2c_1c_2(q-n-1)k^2,$ we get

When $k\neq 0,$ we have

by (4.34), we get $b_1 = b_2 = b_3 = b_4 = b_5 = 0,$ then $c_1 = c_2 = 0,$ so this is a contradiction.

When $k = 0,$ we get $q-n+2 = 0,$ $c_0 = 0$ and $h = c_1+c_2 = c^*.$

5.   Conclusions

For Riemannian supergeometry, we give some definitions about a semi-symmetric non-metric connection. Then by computations, we get the curvature tensor $R_{\widehat{\nabla}^{\mathbb{R}}}$ and the Ricci tensor ${\rm Ric}^{\widehat{\nabla}^\mu}$ of a semi-symmetric non-metric connection on super warped product spaces respectively. We find that they are different from Riemannian geometry. Next, we construct an Einstein super warped product with a semi-symmetric non-metric connection and give another example. The main results of this paper are Theorems 4.3–4.8, which are the conditions that two super warped product spaces with a semi-symmetric non-metric connection are the Einstein super spaces with a semi-symmetric non-metric connection. Moreover, some properties of a semi-symmetric non-metric connection on super warped product spaces are discussed in this paper.

In the future, we can do more research on super Riemannian manifolds.

Acknowledgments

The second author was supported in part by NSFC (No.11771070).

Conflict of interest

The authors declare no conflict of interest.