Pectively. For that reason, from (3), we conclude that (1) normally holds and (2) holds if
Pectively. Therefore, from (three), we conclude that (1) often holds and (2) holds if and only if R N (, ) c2 and 0 c2 , that may be to say, the sectional curvature of N n1-k is bounded from beneath. k However, by (Remark 0.26, [21]), R1 N n1-k is locally symmetric if and k and N n1-k are locally symmetric manifolds, which confirms our claim. only if each Rk 1 Instance 2. Let (F1 ( cn ), g F ) (k = 1) be a Lorentz space type with the constant sectional curvature c1 n1-k , g ) be a Riemannian manifold. We take into account the semi-Riemannian warped N n and ( N solution Benidipine Epigenetic Reader Domain manifold k c F1 ( 1 ) f N n 1 – k n k 1 with the metric g = g F f two g N , exactly where f 0 can be a smooth function defined on F1 ( cn ). Then, the warped product manifold is an Einstein manifold (together with the continuous c1 ) satisfying (1) and (two) if and only if:(i)The Hessian H f in the function f satisfies Hf c = 1 gF , f n (four)Mathematics 2021, 9,three of(ii)N n1-k is definitely an Einstein manifold with its Ricci tensor satisfying Ric N = (n – k) along with the sectional curvature satisfying R N (, ) c2 f two g F ( for any linear independent vector fields , ; f, f) (six) c1 two f gF ( n f, f ) gN , (5)(iii)c1 c2 . n(7)Proof. Firstly, we give two fundamental information. BMS-8 Purity & Documentation Following the notations in Instance 1, the sectional k 1 curvatures of F1 ( cn ) f N n1-k , k = 1, are provided by (see [22], Proposition 4.two) R(u, v) = c1 , n R(u, ) = H f (u, u) , f g F (u, u) R(, ) = R N (, ) – g F ( f2 f, f) , (8)k 1 exactly where u, v on F1 ( cn ) and , on N n1-k are linear independent, respectively. k 1 Moreover, depending on ([23], Corollary three), F1 ( cn ) f N n1-k is an Einstein manifold with Einstein constant c1 if and only if:(a)k 1 The Ricci tensor of F1 ( cn ) satisfiesRic F = c1 g F – (b) N n1-k is an Einstein manifold withn1-k f H , fRic N = N , exactly where can be a continuous offered by = – f f (n – k) gF (( k -1) c1 gF , nf,f ) c1 f two .(9)Now, we assert that (a) with each other with (b) are equivalent to (four) and (five). Considering the fact that Ric F = we understand that (a) is equivalent to (4). Owing to (a), we get f := tr( H f ) = kc1 f , n k 1 and therefore (b) is equivalent to (five). So, we are able to say that F1 ( cn ) f N n1-k is an Einstein manifold together with the continuous c1 if and only if (4) and (5) hold. Then, we prove its sufficiency and necessity. Resulting from the obviousness on the sufficiency k 1 , i.e., if (i), (ii) and (iii) hold, F1 ( cn ) f N n1-k is definitely an Einstein manifold with all the continuous c1 and satisfies (1) and (two), we next prove the necessity . k 1 The two simple information above show that, if F1 ( cn ) f N n1-k is an Einstein manifold using the continual c1 , then (4) and (5) hold and, applying (4), we additional know (8) reduces to R(u, v) = R(u, ) = c1 , n (ten)which suggests (1) is automatic. k 1 Alternatively, considering that F1 ( cn ) f N n1-k satisfies (two), we receive, from (eight) and (10), that R N (, ) – g F ( f , f ) c c2 and 1 c2 , two n f that is, (6) and (7). k 1 To sum up, F1 ( cn ) f N n1-k is an Einstein manifold using the continual c1 and satisfies (1) and (two) if and only if (i), (ii) and (iii) hold.Mathematics 2021, 9,4 ofLet us suppose that k = 1; then, Example 2 becomes the so-called generalized Robertson-Walker spacetime. Then, we’ve got the following Example 3. Example three. We consider the generalized Robertson alker spacetime I f Nn endowed with metric g = -dt2 f 2 (t) g N , exactly where I R is an open interval and f : I R is a smooth function. Then, the generalized Robertson alker spacetime is definitely an Einstein manifold satisfying (1) and (two) if and only if (i) N n is definitely an Einstein manifold together with the continual N and its sec.
