Noncommutative minimal embeddings and morphisms of pseudo-Riemannian calculi
aa r X i v : . [ m a t h . QA ] J un NONCOMMUTATIVE MINIMAL EMBEDDINGS ANDMORPHISMS OF PSEUDO-RIEMANNIAN CALCULI
JOAKIM ARNLIND AND AXEL TIGER NORKVIST
Abstract.
In analogy with classical submanifold theory, we introduce morphismsof real metric calculi together with noncommutative embeddings. We show thatbasic concepts, such as the second fundamental form and the Weingarten map,translate into the noncommutative setting and, in particular, we prove a non-commutative analogue of Gauss’ equations for the curvature of a submanifold.Moreover, the mean curvature of an embedding is readily introduced, giving anatural definition of a noncommutative minimal embedding, and we illustrate thenovel concepts by considering the noncommutative torus as a minimal surface inthe noncommutative 3-sphere. Introduction
In recent years, a lot of progress has been made in understanding the Riemannianaspects of noncommutative geometry. The Levi-Civita connection of a metric playsa crucial role in classical Riemannian geometry and it is important to understand towhat extent a corresponding noncommutative theory exists. Several impressive resultsexist, which computes the curvature of the noncommutative torus from the heat kernelexpansion and consider analogues of the classical Gauss-Bonnet theorem [CT11, FK12,FK13, CM14]. However, starting from a spectral triple, with the metric implicitly givenvia the Dirac operator, it is far from obvious if there exists a module together witha bilinear form, representing the metric corresponding to the Dirac operator, not tomention the existence of a Levi-Civita connection. In order to better understand whatkind of results one can expect, it is interesting to take a more naive approach, whereone starts with a module together with a metric, and tries to understand under whatconditions one may discuss metric compatibility, as well as torsion and uniqueness, ofa general connection.In [AW17a, AW17b], psuedo-Riemannian calculi were introduced as a framework todiscuss the existence of a metric and torsion free connection as well as properties ofits curvature. In fact, the theory is somewhat similar to that of Lie-Rinehart algebras,where a real calculus (as introduced in [AW17b]) might be considered as a “noncom-mutative Lie-Rinehart algebra”. Lie-Rinehart algebras have been discussed from manypoints of view (see e.g. [Rin63, Hue90] and [AAS19] for an overview of metric aspects).Although the existence of a Levi-Civita connection is not always guaranteed in thecontext of pseudo-Rimannian calculi, it was shown that the connection is unique if itexists. The theory has concrete similarities with classical differential geometry, andseveral ideas, such as Koszul’s formula, have direct analogues in the noncommutative
Mathematics Subject Classification. setting. Apart from the noncommutative torus, noncommutative spheres were consid-ered, and a Chern-Gauss-Bonnet type theorem was proven for the noncommutative4-sphere [AW17a]. Note that there are several approaches to metric aspects of non-commutative geometry, and Levi-Civita connections, which are different but similar inspirit (see e.g. [LM87, FGR99, BM11, Ros13, MW18]).In this paper, we introduce morphisms of real (metric) calculi and define noncom-mutative (isometric) embeddings. We show that several basic concepts of submanifoldtheory extends to noncommutative submanifolds and we prove an analogue of Gauss’equations for the curvature of a submanifold. Moreover, the mean curvature of anembedding is defined, immediately giving a natural definition of a (noncommutative)minimal embedding. As an illustration of the above concepts, the noncommutativetorus is considered as a minimal submanifold of the noncommutative 3-sphere.2. Pseudo-Riemannian calculi
Let us briefly recall the basic definitions leading to the concept of a pseudo-Riemanniancalculus and the uniqueness of the Levi-Civita connection. For more details, we referto [AW17b].
Definition 2.1 (Real calculus) . Let A be a unital ∗ -algebra, let g ⊆ Der( A ) be a finite-dimensional (real) Lie algebra over A and let M be a (right) A -module. Moreover, let ϕ : g → M be a R -linear map whose image generates M as an A -module. Then C A = ( A , g , M, ϕ ) is called a real calculus over A .The motivation for the above definition comes from the analogous structures in differ-ential geometry, as seen in the following example. Example 2.2.
Let Σ be a smooth manifold. Then Σ can be represented by the realcalculus C A = ( A , g , M, ϕ ) with A = C ∞ (Σ) , g = Der( C ∞ (Σ)) , M = Vect( M ) (themodule of vector fields on Σ ) and choosing ϕ to be the natural isomorphism betweenthe set of derivations of C ∞ (Σ) and smooth vector fields on Σ . Next, since we are interested in Riemannian geometry, one introduces a metric structureon the module M . Definition 2.3.
Suppose that A is a ∗ -algebra and let M be a right A -module. A hermitian form on M is a map h : M × M → A with the following properties: h h ( m , m + m ) = h ( m , m ) + h ( m , m ) h h ( m , m a ) = h ( m , m ) ah h ( m , m ) = h ( m , m ) ∗ for all m , m , m ∈ M and a ∈ A . Moreover, if h ( m , m ) = 0 for all m ∈ M impliesthat m = 0 then h is said to be nondegenerate , and in this case we say that h is a metric on M . The pair ( M, h ) is called a (right) hermitian A - module , and if h is ametric on M we say that ( M, h ) is a (right) metric A - module . Definition 2.4 (Real metric calculus) . Suppose that C A = ( A , g , M, ϕ ) is a realcalculus over A and that ( M, h ) is a (right) metric A -module. If h ( ϕ ( ∂ ) , ϕ ( ∂ )) ∗ = h ( ϕ ( ∂ ) , ϕ ( ∂ ))for all ∂ , ∂ ∈ g then the pair ( C A , h ) is called a real metric calculus . Example 2.5.
Let (Σ , g ) be a Riemannian manifold and let C A be the real calculusfrom Example 2.2 representing Σ . Then ( C A , g ) is a real metric calculus. In what follows, we shall sometimes require the metric to satisfy a stronger conditionthan nondegeneracy.
Definition 2.6.
Let h be a metric on M and let ˆ h : M → M ∗ (the dual of M ) bethe mapping given by ˆ h ( m )( n ) = h ( m, n ). The metric h is said to be invertible if ˆ h isinvertible.Now, given a real metric calculus C A = ( A , g , M, ϕ ), we will discuss connections on M and their compatibility with the metric. Let us start by recalling the definition of anaffine connection for a derivation based calculus. Definition 2.7.
Let C A = ( A , g , M, ϕ ) be a real calculus over A . An affine connection on ( M, g ) is a map ∇ : g × M → M satisfying(1) ∇ ∂ ( m + n ) = ∇ ∂ m + ∇ ∂ n ,(2) ∇ λ∂ + ∂ ′ m = λ ∇ ∂ m + ∇ ∂ ′ m ,(3) ∇ ∂ ( ma ) = ( ∇ ∂ m ) a + m∂ ( a )for m, n ∈ M , ∂, ∂ ′ ∈ g , a ∈ A and λ ∈ R .The fact that we shall require the connection to be “real” is reflected in the followingdefinition. Definition 2.8.
Let ( C A , h ) be a real metric calculus and let ∇ denote an affineconnection on ( M, g ). Then ( C A , h, ∇ ) is called a real connection calculus if h (cid:0) ∇ ∂ ϕ ( ∂ ) , ϕ ( ∂ ) (cid:1) = h ( ∇ ∂ ϕ ( ∂ ) , ϕ ( ∂ )) ∗ for all ∂, ∂ , ∂ ∈ g . Definition 2.9.
Let ( C A , h, ∇ ) be a real connection calculus. We say that ( C A , h, ∇ )is metric if ∂ ( h ( m, n )) = h ( ∇ ∂ m, n ) + h ( m, ∇ ∂ n )for all ∂ ∈ g and m, n ∈ M , and torsion-free if ∇ ∂ ϕ ( ∂ ) − ∇ ∂ ϕ ( ∂ ) − ϕ ([ ∂ , ∂ ]) = 0for all ∂ , ∂ ∈ g . A metric and torsion-free real connection calculus is called a pseudo-Riemannian calculus .A connection fulfilling the requirements of a pseudo-Riemannian calculus is called a Levi-Civita connection . In the quite general setup of real metric calculi, where thereare few assumptions on the structure of the algebra A and the module M , the existenceof a Levi-Civita connection can not be guaranteed. However, if it exists, it is unique. Theorem 2.10. ([AW17b])
Let ( C A , h ) be a real metric calculus. Then there exists atmost one affine connection ∇ such that ( C A , h, ∇ ) is a pseudo-Riemannian calculus. The next result provides us a noncommutative analogue of Koszul’s formula, which isa useful tool for constructing the Levi-Civita connection in several examples.
Proposition 2.11. ([AW17b])
Let ( C A , h, ∇ ) be a pseudo-Riemannian calculus andassume that ∂ , ∂ , ∂ ∈ g . Then it holds that (2.1) 2 h ( ∇ E , E ) = ∂ h ( E , E ) + ∂ h ( E , E ) − ∂ h ( E , E ) − h ( E , ϕ ([ ∂ , ∂ ])) + h ( E , ϕ ([ ∂ , ∂ ])) + h ( E , ϕ ([ ∂ , ∂ ])) , where ∇ i = ∇ ∂ i and E i = ϕ ( ∂ i ) for i = 1 , , . As in Riemannian geometry, a connection satisfying Koszul’s formula is torsion-freeand compatible with the metric.
Proposition 2.12. ([AW17b])
Let ( C A , h ) be a real metric calculus, and suppose that ∇ is an affine connection on ( M, g ) such that Koszul’s formula (2.1) holds. Then ( C A , h, ∇ ) is a pseudo-Riemannian calculus. A particularly simple case, which is also relevant to our applications, is when M isa free module. The following result then gives a way of constructing the Levi-Civitaconnection from Koszul’s formula. Corollary 2.13. ([AW17b])
Let ( C A , h ) be a real metric calculus and let { ∂ , ..., ∂ n } be a basis of g such that { E a = ϕ ( ∂ a ) } na =1 is a basis for M . If there exist m ab ∈ M such that (2.2) h ( m ab , E c ) = ∂ a h ( E b , E c ) + ∂ b h ( E a , E c ) − ∂ c h ( E a , E b ) − h ( E a , ϕ ([ ∂ b , ∂ c ])) + h ( E b , ϕ ([ ∂ c , ∂ a ])) + h ( E c , ϕ ([ ∂ a , ∂ b ])) , for a, b, c = 1 , ..., n , then there exists an affine connection ∇ , given by ∇ ∂ a E b = m ab ,such that ( C A , h, ∇ ) is a pseudo-Riemannian calculus. Real calculus homomorphisms
In order to understand the algebraic structure of real calculi, a first step is to considermorphisms. Via a concept of morphism of real calculi, one can understand when twocalculi are considered to be equal (isomorphic) and, from a geometric point of view,it opens up for noncommutative embeddings. In this section we introduce homomor-phisms of real (metric) calculi and prove several results which, in different ways, shedlight on the new concept.
Definition 3.1.
Let C A = ( A , g , M, ϕ ) and C A ′ = ( A ′ , g ′ , M ′ , ϕ ′ ) be real caculi andassume that φ : A → A ′ is a ∗ -algebra homomorphism. If there is a map ψ : g ′ → g such that( ψ ψ is a Lie algebra homomorphism( ψ δ ( φ ( a )) = φ ( ψ ( δ )( a )) for all δ ∈ g ′ , a ∈ A ,then ψ is said to be compatible with φ . If ψ is compatible with φ we define Ψ asΨ = ϕ ◦ ψ , and M Ψ is defined to be the submodule of M generated by Ψ( g ′ ).Furthermore, if there is a map b ψ : M Ψ → M ′ such that( b ψ b ψ ( m + m ) = b ψ ( m ) + b ψ ( m ) for all m , m ∈ M ( b ψ b ψ ( ma ) = b ψ ( m ) φ ( a ) for all m ∈ M and a ∈ A ( b ψ b ψ (Ψ( δ )) = ϕ ′ ( δ ) for all δ ∈ g ′ , then b ψ is said to be compatible with φ and ψ , and ( φ, ψ, b ψ ) is called a real calculushomomorphism from C A to C A ′ (see Figure 1 for an illustration of a real calculushomomorphism). If φ is a ∗ -algebra isomorphism, ψ a Lie algebra isomorphism and b ψ is a bijective map then ( φ, ψ, b ψ ) is called a real calculus isomorphism. C A C A ′ A g ψ ( g ′ ) MM Ψ A ′ g ′ M ′ φψ b ψϕ ϕ ′ Figure 1.
A real calculus homomorphism ( φ, ψ, b ψ ) : C A → C A ′ .Let us try to understand Definition 3.1 in the context of embeddings, where theanalogy with classical geometry is rather clear. Thus, let φ : Σ ′ → Σ be an embeddingof Σ ′ into Σ and let φ : C ∞ (Σ) → C ∞ (Σ ′ ) be the corresponding homomorphism of thealgebras of smooth functions. In the notation of Definition 3.1 we have A = C ∞ (Σ) φ −→ A ′ = C ∞ (Σ ′ ) g = Der( A ) ψ ←− g ′ = Der( A ′ ) M = Vect(Σ) ⊇ M Ψ b ψ −→ M ′ = Vect(Σ ′ ) . First of all, there is no natural map from Vect(Σ) to Vect(Σ ′ ) since a vector field X ∈ Vect(Σ) at a point p ∈ φ (Σ ′ ) might not lie in T p Σ ′ (regarded as a subspace of T p Σ). However, vector fields which are tangent to Σ ′ in this sense may be restricted toΣ ′ . On the other hand, any vector field X ′ ∈ Vect(Σ ′ ) (assuming Σ ′ to be closed) canbe extended to a smooth vector field X ∈ Vect(Σ) such that X | Σ ′ = X ′ . In light of theisomorphism between vector fields and derivations, it is therefore more natural to havea map ψ : Der( A ′ ) → Der( A ), corresponding to a choice of extension of vector fieldson Σ ′ . The map b ψ then corresponds to the restriction of vector fields on Σ which aretangent to Σ ′ . Consequently, we consider vector fields in M Ψ as extensions of vectorfields on the embedded manifold.In noncommutative geometry (in contrast to the classical case) g is no longer a A -module, a difference which is captured by the concept of a real calculus. The definitionof homomorphism reflects this fact by assuming that every derivation of A ′ can be “extended” to a derivation of A and, furthermore, that every vector field on Σ whichis tangent to Σ ′ (that is, in the image of ϕ ◦ ψ ) can be “restricted” to Σ ′ .Next, one can easily check that the composition of two homomorphisms is again ahomomorphism. Proposition 3.2.
Let C A , C A ′ and C A ′′ be real calculi and assume that ( φ, ψ, b ψ ) : C A → C A ′ and ( φ ′ , ψ ′ , b ψ ′ ) : C A ′ → C A ′′ are real calculus homomorphisms. Then ( φ ′ ◦ φ, ψ ◦ ψ ′ , b ψ ′ ◦ b ψ ) : C A → C A ′′ is a realcalculus homomorphism.Proof. For convenience, we introduce Φ := φ ′ ◦ φ , ˜ ψ := ψ ◦ ψ ′ and ˆΨ := b ψ ′ ◦ b ψ .First of all, it is clear that Φ is a ∗ -algebra homomorphism and ˜ ψ is a Lie algebrahomomorphism. For a ∈ A and δ ∈ g ′′ we get that δ (Φ( a )) = φ ′ ( ψ ′ ( δ )( φ ( a ))) = φ ′ ( φ ( ˜ ψ ( δ )( a ))) = Φ( ˜ ψ ( δ )( a )) , showing that Φ and ˜ ψ are compatible, with M ˜Ψ being the submodule of M generatedby ˜ ψ ( g ′′ ). Checking that ˆΨ( m + n ) = ˆΨ( m ) + ˆΨ( n ) and ˆΨ( ma ) = ˆΨ( m )Φ( a ) for all m, n ∈ M ˜Ψ and a ∈ A is trivial, and for δ ∈ g ′′ we get ϕ ′′ ( δ ) = b ψ ′ (Ψ ′ ( δ )) = b ψ ′ ( ϕ ′ ( ψ ′ ( δ ))) = b ψ ′ ( b ψ (Ψ( ψ ′ ( δ )))) = ˆΨ( ϕ ◦ ˜ ψ ( δ )) . Thus ˆΨ is compatible with Φ and ˜ ψ , and it follows that (Φ , ˜ ψ, ˆΨ) is a real calculushomomorphism from C A to C A ′′ . (cid:3) A homomorphism of real calculi ( φ, ψ, b ψ ) consists of three maps, and a natural questionis what kind of freedom one has in choosing these maps? Let us start by showingthat, given φ and ψ , there is at most one b ψ such that ( φ, ψ, b ψ ) is a real calculushomomorphism. Proposition 3.3. If ( φ, ψ, b ψ ) and ( φ, ψ, ˜ ψ ) are real calculus homomorphisms from C A to C A ′ then b ψ = ˜ ψ .Proof. Let m = Ψ( δ i ) a i for δ i ∈ g ′ and a i ∈ A be an arbitrary element of M Ψ . Itfollows from ( b ψ b ψ
3) that˜ ψ ( m ) = ˜ ψ (Ψ( δ i ) a i ) = ˜ ψ (Ψ( δ i )) φ ( a i ) = ϕ ′ ( δ i ) φ ( a i ) = b ψ (Ψ( δ i )) φ ( a i )= b ψ (Ψ( δ i ) a i ) = b ψ ( m ) . (cid:3) Furthermore, if φ is an isomorphism, then the next result shows that ψ is determineduniquely by φ . Thus, combined with the previous result we conclude that if ( φ, ψ, b ψ )is an isomorphism of real caluli, then ψ and b ψ are uniquely determined by φ . Proposition 3.4. If ( φ, ψ, b ψ ) : C A → C A ′ is a real calculus homomorphism such that φ is an isomorphism, then ψ is a Lie algebra isomorphism with ψ ( δ ) = φ − ◦ δ ◦ φ for δ ∈ g ′ . Proof.
The formula for ψ follows directly from the fact that δ ( φ ( a )) = φ ( ψ ( δ )( a ))together with φ being an isomorphism. To prove that ψ is an isomorphism, let ˜ ψ : g → g ′ be given by ˜ ψ ( ∂ ) = φ ◦ ∂ ◦ φ − . Then for any ∂ ∈ g and δ ∈ g ′ it follows that ψ ◦ ˜ ψ ( ∂ ) = φ − ◦ ˜ ψ ( ∂ ) ◦ φ = φ − ◦ φ ◦ ∂ ◦ φ − ◦ φ = ∂ ˜ ψ ◦ ψ ( δ ) = φ ◦ ψ ( δ ) ◦ φ − = φ ◦ φ − ◦ δ ◦ φ ◦ φ − = δ. Thus ψ is a bijection with inverse ψ − = ˜ ψ . Furthermore, ψ − preserves the Liebracket: ψ − ([ ∂ , ∂ ]) = ψ − ([ ψ ◦ ψ − ( ∂ ) , ψ ◦ ψ − ( ∂ )]) = ψ − ◦ ψ ([ ψ − ( ∂ ) , ψ − ( ∂ )])= [ ψ − ( ∂ ) , ψ − ( ∂ )] , proving that ψ is indeed a Lie algebra isomorphism. (cid:3) Given a homomorphism ( φ, ψ, b ψ ) : C A → C A ′ , there is a natural A -module structureon M ′ given by m ′ · a = m ′ φ ( a ) for m ′ ∈ M ′ and a ∈ A . As expected, the right A -modules M and M ′ are isomorphic when ( φ, ψ, b ψ ) is an isomorphism. Proposition 3.5. If ( φ, ψ, b ψ ) : C A → C A ′ is a real calculus isomorphism then M = M Ψ ≃ M ′ . Proof.
Since ψ is an isomorphism it follows that g = ψ ( g ′ ). From this it immediatelyfollows that M = M Ψ , since M Ψ is defined to be the submodule of M generated by g = ψ ( g ′ ). Considering M ′ as a right A -module, b ψ is an A -module homomorphism,and since b ψ is assumed to be bijective, we conclude that M Ψ ≃ M ′ . (cid:3) Recalling our previous discussions of real calculus homomorphisms in relation to em-beddings, one may consider vector fields in M Ψ as extensions of vector fields in M . Letus therefore make the following definition. Definition 3.6. If m ∈ M Ψ such that b ψ ( m ) = m ′ then m is called an extension of m ′ .The set of extensions of m ′ will be denoted by Ext Ψ ( m ′ ).3.1. Homomorphisms of real metric calculi.
Having introduced the concept ofhomomorphisms for real calculi, it is natural to proceed to real metric calculi. Fromthe geometric point of view, in the case of embeddings, one would like a homomorphismof real metric calculi to correspond to an isometric embedding. The following definitionis straight-forward.
Definition 3.7.
Let ( C A , h ) and ( C A ′ , h ′ ) be real metric calculi and assume that( φ, ψ, b ψ ) : C A → C A ′ is a real calculus homomorphism. If h ′ (cid:0) ϕ ′ ( δ ) , ϕ ′ ( δ ) (cid:1) = φ (cid:0) h (Ψ( δ ) , Ψ( δ )) (cid:1) for all δ , δ ∈ g ′ then ( φ, ψ, b ψ ) is called a real metric calculus homomorphism .Assume that ( φ, ψ, b ψ ) : ( C A , h ) → C A is a homomorphism of real calculi. It is naturalto ask if there exists a metric h ′ such that ( φ, ψ, b ψ ) : ( C A , h ) → ( C A , h ′ ) is a homo-morphism of real metric calculi, in which case we would call h ′ the induced metric. Asit turns out, one can not guarantee the existence of h ′ , but whenever it exists, it isunique; we state this as follows. Proposition 3.8.
Let C A be a real calculus, ( C A , h ) a real metric calculus, and let ( φ, ψ, b ψ ) : ( C A , h ) → C A ′ be a real calculus homomorphism. Then there exists at mostone hermitian form h ′ on M ′ satisfying h ′ ( ϕ ′ ( δ ) , ϕ ′ ( δ )) = φ (cid:0) h (Ψ( δ ) , Ψ( δ )) (cid:1) , δ , δ ∈ g ′ . Proof.
Suppose that h ′ and h ′ both fulfill the given conditions for h ′ . By definition ofreal calculus homomorphism it is immediately obvious that h ′ and h ′ agree on ϕ ′ ( g ′ ).If we take two arbitrary elements m, n ∈ M ′ it follows from the fact that C A ′ is a realcalculus over A ′ that m and n can be written as m ′ = ϕ ′ ( δ i ) a i , δ i ∈ g ′ , a i ∈ A ′ ,n ′ = ϕ ′ ( δ j ) b j , δ j ∈ g ′ , b j ∈ A ′ . Furthermore, one obtains h ′ ( m ′ , n ′ ) = h ′ (cid:0) ϕ ′ ( δ i ) a i , ϕ ′ ( δ j ) b j (cid:1) = h ′ (cid:0) ϕ ′ ( δ i ) a i , ϕ ′ ( δ j ) (cid:1) b j = ( a i ) ∗ h ′ ( ϕ ′ ( δ i ) , ϕ ′ ( δ j )) b j = ( a i ) ∗ h ′ ( ϕ ′ ( δ i ) , ϕ ′ ( δ j )) b j = h ′ (cid:0) ϕ ′ ( δ i ) a i , ϕ ′ ( δ j ) (cid:1) b j = h ′ (cid:0) ϕ ′ ( δ i ) a i , ϕ ′ ( δ j ) b j (cid:1) = h ′ ( m ′ , n ′ ) , since h ′ and h ′ are hermitian forms on M ′ and h ′ ( ϕ ′ ( δ i ) , ϕ ′ ( δ j )) = h ′ ( ϕ ′ ( δ i ) , ϕ ′ ( δ j ))for δ , δ ∈ g ′ . Since m ′ and n ′ are arbitrary, it follows that h ′ = h ′ . (cid:3) Note that if ( φ, ψ, b ψ ) : ( C A , h ) → ( C A ′ , h ′ ) is a homomorphism of real metric calculi,then φ (cid:0) h ( m, n ) (cid:1) = h ′ ( b ψ ( m ) , b ψ ( n )) for all m, n ∈ M Ψ . In other words φ (cid:0) h ( m, n ) (cid:1) = h ′ ( m ′ , n ′ )if m ∈ Ext Ψ ( m ′ ) and n ∈ Ext Ψ ( n ′ ). This is to be compared with the geometrical situ-ation where the inner product of vector fields restricted to the isometrically embeddedmanifolds equals the inner product of the restricted vector fields.4. Embeddings of real caluli
In the previous section, we highlighted the analogy with embedded manifolds in orderto motivate and understand the different concepts introduced for noncommutativealgebras. However, we did not make the distinction between general homomorphismsand embeddings precise. In this section we shall define noncommutative embeddingsand introduce a theory of submanifolds, much in analogy with the classical situation.It turns out that one can readily introduce the second fundamental form, and find anoncommutative analogue of Gauss’ equation, giving the curvature of the submanifold.A necessary condition for a map φ : Σ ′ → Σ to be an embedding, is that φ isinjective; dually, this corresponds to φ : C ∞ (Σ) → C ∞ (Σ ′ ) being surjective. To for-mulate the next definition, we recall the orthogonal complement of a module. Namely,let ( C A , h ) be a real metric calculus. Given any subset N ⊆ M , we define N ⊥ = { m ∈ M : h ( m, n ) = 0 } and note that N ⊥ is a A -module. Definition 4.1.
A homomorphism of real calculi ( φ, ψ, b ψ ) : C A → C A ′ is called an embedding if φ is surjective and there exists a submodule ˜ M ⊆ M such that M = M Ψ ⊕ ˜ M . A homomorphism of real metric calculi ( φ, ψ, b ψ ) : ( C A , h ) → ( C A ′ , h ′ ) iscalled an isometric embedding if ( φ, ψ, b ψ ) is an embedding and M = M Ψ ⊕ M ⊥ Ψ . The surjectivity of φ has immediate implications for the maps ψ and b ψ . Proposition 4.2.
Assume that ( φ, ψ, b ψ ) : C A → C A ′ is a real calculus homomorphismsuch that that φ is surjective. Then ψ is injective and b ψ is surjective.Proof. For the first statement, suppose δ ∈ ker( ψ ). Then for any a ∈ A it follows that ψ ( δ )( a ) = 0. Thus, by ( ψ
2) it follows that δ ( φ ( a )) = φ ( ψ ( δ )( a )) = φ (0) = 0for any a ∈ A , and since φ is surjective it follows that δ ( a ′ ) = 0 for every a ′ ∈ A ′ .For the second statement, let m ′ ∈ M ′ . Then m ′ can be written on the form m ′ = ϕ ′ ( δ i ) b i for some δ i ∈ g ′ and b i ∈ A ′ , and since φ is surjective there are a i ∈ A such that φ ( a i ) = b i . It follows that m ′ = ϕ ′ ( δ i ) b i = b ψ (Ψ( δ i )) φ ( a i ) = b ψ (cid:0) Ψ( δ i ) a i (cid:1) , completing the proof. (cid:3) Note that Proposition 4.2 gives further motivation for Definition 4.1 since it showsthat ψ is injective, in analogy with the injectivity of the tangent map of an embedding.Moreover, it follows from Proposition 4.2 that if ( φ, ψ, b ψ ) : C A → C A ′ is an embedding,then every element m ′ ∈ M ′ has at least one extension corresponding to the geometricsituation where a vector field on the embedded manifold can be extended to a vectorfield in the ambient space.Furthermore, given an embedding ( φ, ψ, b ψ ) : C A → C A ′ , we define the A -linearprojection P : M → M Ψ as P ( m Ψ ⊕ ˜ m ) = m Ψ with respect to the decomposition M = M Ψ ⊕ ˜ M . The complementary projection willbe denoted by Π = − P . (Note that for an embedding of real metric calculi, theprojections P and Π are orthogonal with respect to the metric on M .)In analogy with classical Riemannian submanifold theory (see e.g. [KN96]), one de-composes the Levi-Civita connection in its tangential and normal parts. Let ( C A , h, ∇ )and ( C A ′ , h ′ , ∇ ′ ) be psuedo-Riemannian calculi and assume that ( φ, ψ, b ψ ) : ( C A , h ) → ( C A ′ , h ′ ) is an isometric embedding and write ∇ ψ ( δ ) m = L ( δ, m ) + α ( δ, m )(4.1) ∇ ψ ( δ ) ξ = − A ξ ( δ ) + D δ ξ (4.2)for δ ∈ g ′ , m ∈ M Ψ and ξ ∈ M ⊥ Ψ , with L ( δ, m ) = P ( ∇ ψ ( δ ) m ) α ( δ, m ) = Π( ∇ ψ ( δ ) m ) A ξ ( δ ) = − P ( ∇ ψ ( δ ) ξ ) D δ ξ = Π( ∇ ψ ( δ ) ξ ) . In differential geometry, (4.1) is called
Gauss’ forumla and (4.2) is called
Weingarten’sformula . Furthermore, α : g ′ × M Ψ → M ⊥ Ψ is called the second fundamental form and A : g ′ × M ⊥ Ψ → M Ψ is called the Weingarten map . Let us start by showing that thetangential part L ( δ, m ) is an extension of the Levi-Civita connection on ( C A ′ , h ′ , ∇ ′ ). Proposition 4.3. If δ ∈ g ′ and m ∈ Ext Ψ ( m ′ ) then L ( δ, m ) ∈ Ext Ψ ( ∇ ′ δ m ′ ) Proof.
For the sake of readability, let us first establish some notation. Let δ i ∈ g ′ andlet ∂ i = ψ ( δ i ) , E i = Ψ( δ i ) and E ′ i = ϕ ′ ( δ i ). Moreover, let h ij = h ( E i , E j ) and let h i, [ j,k ] = h ( E i , Ψ([ δ j , δ k ])); likewise, let h ′ ij = h ( E ′ i , E ′ j ) and h ′ i, [ j,k ] = h ( E ′ i , ϕ ′ ([ δ j , δ k ])).With this notation in place, Koszul’s formula yields2 h ( ∇ i E j , E k ) = ∂ i h jk + ∂ j h ik − ∂ k h ij − h i, [ j,k ] + h j, [ k,i ] + h k, [ i,j ] h ′ ( ∇ ′ i E ′ j , E ′ k ) = δ i h ′ jk + δ j h ′ ik − δ k h ′ ij − h ′ i, [ j,k ] + h ′ j, [ k,i ] + h ′ k, [ i,j ] for all δ i , δ j , δ k ∈ g ′ , and since h ′ is induced from h it follows that h ′ jk = φ ( h jk ) h ′ i, [ j,k ] = φ ( h i, [ j,k ] ) δ i h ′ jk = δ i φ ( h jk ) = φ ( ∂ i ( h jk ));from this it becomes clear that h ′ ( ∇ ′ i E ′ j , E ′ k ) = φ ( h ( ∇ i E j , E k )). Let m = E i a i ∈ M Ψ and n = E k b k ∈ M Ψ be arbitrary elements in M Ψ , where a i , b k ∈ A . By definition ofaffine connections it follows that h ( ∇ j m, n ) = h ( ∇ j ( E i a i ) , E k b k ) = h (( ∇ j E i ) a i , E k b k ) + h ( E i ∂ j ( a i ) , E k b k )= ( a i ) ∗ h ( ∇ j E i , E k ) b k + ∂ j ( a i ) ∗ h ik b k , and we get φ ( h ( ∇ j m, n )) = φ ( a i ) ∗ h ′ ( ∇ ′ j E ′ i , E ′ k ) φ ( b k ) + φ ( ∂ j ( a i ) ∗ ) h ′ ik φ ( b k )= φ ( a i ) ∗ h ′ ( ∇ ′ j E ′ i , E ′ k ) φ ( b k ) + δ j ( φ ( a i ) ∗ ) h ′ ik φ ( b k )= h ′ (( ∇ ′ j E ′ i ) φ ( a i ) , E ′ k φ ( b k )) + h ′ ( E ′ i δ j ( φ ( a i )) , E ′ k φ ( b k ))= h ′ ( ∇ ′ j ( E ′ i φ ( a i )) , E ′ k φ ( b k )) = h ′ ( ∇ ′ j ( b ψ ( m )) , b ψ ( n )) . It now follows that h ′ ( ∇ ′ j ( b ψ ( m )) , b ψ ( n )) = φ ( h ( ∇ j m, n )) = φ ( h ( P ( ∇ j m ) , n )) = φ ( h ( L ( δ j , m ) , n )) , which equals h ′ ( b ψ ( L ( δ j , m )) , b ψ ( n )). Thus, h ′ ( ∇ ′ j ( b ψ ( m )) , b ψ ( n )) = h ′ ( b ψ ( L ( δ j , m )) , b ψ ( n )) , and since h ′ is nondegenerate and b ψ is surjective, it follows that b ψ ( L ( δ j , m )) = ∇ ′ j b ψ ( m )which is equivalent to L ( δ j , m ) ∈ Ext Ψ ( ∇ ′ j b ψ ( m )), and it immediately follows that if m ∈ Ext Ψ ( m ′ ) then L ( δ, m ) ∈ Ext Ψ ( ∇ δ m ′ ) for any δ ∈ g ′ and m ′ ∈ M ′ . (cid:3) In view of the above result, we introduce the notation L ( δ, m ) = ˆ ∇ ′ δ m and concludethat ∇ ′ δ m ′ = b ψ (cid:0) ˆ ∇ ′ δ m (cid:1) = b ψ (cid:0) P ( ∇ ψ ( δ ) m ) (cid:1) if m ∈ Ext Ψ ( m ′ ), giving a convenient way of retrieving the Levi-Civita connection ∇ ′ from ∇ . Next, let us show that the second fundamental form shares the properties ofits classical counterpart. Proposition 4.4. If δ , δ ∈ g ′ , a , a ∈ A and λ , λ ∈ R then α (cid:0) δ , Ψ( δ ) (cid:1) = α (cid:0) δ , Ψ( δ ) (cid:1) α (cid:0) λ δ + λ δ , m (cid:1) = λ α ( δ , m ) + λ α ( δ , m ) α ( δ , m a + m a ) = α ( δ , m ) a + α ( δ , m ) a for m , m ∈ M Ψ .Proof. For the first statement, let ∆( δ , δ ) = α ( δ , Ψ( δ )) − α ( δ , Ψ( δ )). With thisnotation in place one may use the fact that ∇ is torsion-free to get:0 = ∇ ψ ( δ ) Ψ( δ ) − ∇ ψ ( δ ) Ψ( δ ) − ϕ ([ ψ ( δ ) , ψ ( δ )])= ∇ ψ ( δ ) Ψ( δ ) − ∇ ψ ( δ ) Ψ( δ ) − Ψ([( δ ) , ( δ )])= P ( ∇ ψ ( δ ) Ψ( δ )) − P ( ∇ ψ ( δ ) Ψ( δ )) − Ψ([ δ , δ ]) + ∆( δ , δ ) , and since the projection P is linear, together with the fact that P (Ψ([ δ , δ ])) =Ψ([ δ , δ ]) ∈ M Ψ , it follows that0 = P ( ∇ ψ ( δ ) Ψ( δ ) − ∇ ψ ( δ ) Ψ( δ ) − Ψ([ δ , δ ])) + ∆( δ , δ )= P (0) + ∆( δ , δ ) = 0 + ∆( δ , δ ) = ∆( δ , δ ) . For the second and third statements we use the linearity of the connection: α (cid:0) λ δ + λ δ , m (cid:1) = ( − P ) (cid:0) ∇ ψ ( λ δ + λ δ ) m (cid:1) = ( − P ) (cid:0) λ ∇ ψ ( δ ) m + λ ∇ ψ ( δ ) m (cid:1) = λ α (cid:0) δ , m (cid:1) + λ α (cid:0) δ , m (cid:1) and α ( δ , m a + m a ) = ( − P ) ( ∇ δ ( m a + m a ))= ( − P ) ( ∇ δ ( m a ) + ∇ δ ( m a ))= α ( δ , m a ) + α ( δ , m a ) . Noting that α ( δ , m a ) = ∇ ψ ( δ ) m a − P ( ∇ ψ ( δ ) m a )= ( ∇ ψ ( δ ) m ) a + m ψ ( δ )( a ) − P (cid:0) ( ∇ ψ ( δ ) m ) a + m ψ ( δ )( a ) (cid:1) = ( ∇ ψ ( δ ) m ) a + m ψ ( δ )( a ) − P ( ∇ ψ ( δ ) m ) a − m ψ ( δ )( a )= ( ∇ ψ ( δ ) m − P ( ∇ ψ ( δ ) m )) a = α ( δ , m ) a and (similarly) that α ( δ , m a ) = α ( δ , m ) a the proposition now follows. (cid:3) Proposition 4.5. If δ ∈ g ′ , m ∈ M Ψ and ξ ∈ M ⊥ Ψ then h (cid:0) A ξ ( δ ) , m (cid:1) = h (cid:0) ξ, α ( δ, m ) (cid:1) . Proof.
Since h ( m, ξ ) = 0 one can use that ( C A , h, ∇ ) is metric to see that 0 = ψ ( δ )( h ( m, ξ )) = h ( ∇ ψ ( δ ) ξ, m ) + h ( ξ, ∇ ψ ( δ ) m ). Using that P is an orthogonal pro-jection, it follows that h ( A ξ ( δ ) , m ) = − h ( P ( ∇ ψ ( δ ) ξ ) , m )= − h ( ∇ ψ ( δ ) ξ, m ) = h ( ξ, ∇ ψ ( δ ) m ) = h (cid:0) ξ, α ( δ, m ) (cid:1) as desired. (cid:3) Having considered properties of L , α and A ξ , let us now show that D X has the prop-erties of an affine connection; in differential geometry, D X is usually identified with aconnection on the normal bundle of the submanifold. Proposition 4.6. If δ , δ ∈ g ′ , ξ , ξ ∈ M ⊥ Ψ , λ ∈ R and a ∈ A then (1) D δ ( ξ + ξ ) = D δ ξ + D δ ξ , (2) D λδ + δ ξ = λD δ ξ + D δ ξ , (3) D δ ( ξ a ) = ( D δ ξ ) a + ξ ψ ( δ )( a ) .Proof. Note that (1) and (2) follows immediately from the linearity of ∇ . To prove(3), one computes the left-hand side directly: D δ ( ξ a ) = Π( ∇ ψ ( δ ) ξ a ) = Π(( ∇ ψ ( δ ) ξ ) a + ξ ψ ( δ )( a ))= Π(( ∇ ψ ( δ ) ξ ) a ) + Π( ξ ψ ( δ )( a )) = ( D δ ξ ) a + ξ ψ ( δ )( a ) , giving the desired result. (cid:3) A classical formula in Riemannian geometry is Gauss’ equation, which relates thecurvature of the ambient space to the curvature of the submanifold. The next resultprovides a noncommutative analogue.
Proposition 4.7 (Gauss’ equation) . Let δ i ∈ g ′ , ∂ i = ψ ( δ i ) ∈ g , E i = Ψ( δ i ) ∈ M Ψ and E ′ i = ϕ ′ ( δ i ) ∈ M ′ for i = 1 , , , (i.e. E i is an extension of E ′ i ). Then φ (cid:0) h ( E , R ( ∂ , ∂ ) E ) (cid:1) = h ′ (cid:0) E ′ , R ′ ( δ , δ ) E ′ (cid:1) + φ (cid:0) h (cid:0) α ( δ , E ) , α ( δ , E ) (cid:1)(cid:1) − φ (cid:0) h (cid:0) α ( δ , E ) , α ( δ , E ) (cid:1)(cid:1) . (4.3) Proof.
Using the result from Proposition 4.3 one gets that R ′ ( δ , δ ) E ′ = ∇ ′ ∇ ′ E ′ − ∇ ′ ∇ ′ E ′ − ∇ ′ [ δ ,δ ] E ′ = ∇ ′ b ψ ( ˆ ∇ ′ E ) − ∇ ′ b ψ ( ˆ ∇ ′ E ) − b ψ ( ˆ ∇ ′ [ δ ,δ ] E )= b ψ (cid:16) ˆ ∇ ′ ˆ ∇ ′ E − ˆ ∇ ′ ˆ ∇ ′ E − ˆ ∇ ′ [ δ ,δ ] E (cid:17) . Setting ˆ R ( ∂ , ∂ ) E := ˆ ∇ ′ ˆ ∇ ′ E − ˆ ∇ ′ ˆ ∇ ′ E − ˆ ∇ ′ [ δ ,δ ] E one obtains h ′ (cid:0) E ′ , R ′ ( δ , δ ) E ′ (cid:1) = h ′ (cid:0) b ψ ( E ) , b ψ ( ˆ R ( ∂ , ∂ ) E ) (cid:1) = φ ( h (cid:0) E , ˆ R ( ∂ , ∂ ) E (cid:1) )= φ (cid:16) h (cid:0) E , ˆ ∇ ′ ˆ ∇ ′ E − ˆ ∇ ′ ˆ ∇ ′ E − ˆ ∇ ′ [ δ ,δ ] E (cid:1)(cid:17) = φ (cid:16) h (cid:0) E , ∇ ˆ ∇ ′ E − ∇ ˆ ∇ ′ E − ∇ [ ∂ ,∂ ] E (cid:1)(cid:17) , since E ∈ M Ψ . Using the fact that ∇ i ˆ ∇ ′ j E k = ∇ i ( ∇ j E k − α ( δ j , E k )) one may write h ′ (cid:0) E ′ , R ′ ( δ , δ ) E ′ (cid:1) = φ (cid:0) h (cid:0) E , R ( ∂ , ∂ ) E − ∇ α ( δ , E ) + ∇ α ( δ , E ) (cid:1)(cid:1) , and from this it follows immediately that φ (cid:0) h ( E , R ( ∂ , ∂ ) E ) (cid:1) = h ′ (cid:0) E ′ , R ′ ( δ , δ ) E ′ (cid:1) + φ (cid:0) h (cid:0) E , ∇ α ( δ , E ) (cid:1)(cid:1) − φ (cid:0) h (cid:0) E , ∇ α ( δ , E ) (cid:1)(cid:1) . Since ( C A , h, ∇ ) is metric it follows that h ( E , ∇ ψ ( δ ) ξ ) = − h ( ∇ ψ ( δ ) E , ξ )for ξ ∈ M ⊥ Ψ , implying that φ (cid:0) h ( E , R ( ∂ , ∂ ) E ) (cid:1) = h ′ (cid:0) E ′ , R ′ ( δ , δ ) E ′ (cid:1) + φ (cid:0) h (cid:0) ∇ E , α ( δ , E ) (cid:1)(cid:1) − φ (cid:0) h (cid:0) ∇ E , α ( δ , E ) (cid:1)(cid:1) , which completes the proof, since h (cid:0) ∇ E , α ( δ , E ) (cid:1) = h (cid:0) α ( δ , E ) , α ( δ , E ) (cid:1) and h (cid:0) ∇ E , α ( δ , E ) (cid:1) = h (cid:0) α ( δ , E ) , α ( δ , E ) (cid:1) . (cid:3) Free real calculi and noncommutative mean curvature
In the examples we shall consider (the noncommutative torus and the noncommutative3-sphere), M will be a free module with a basis given by the image of a basis ofthe Lie algebra g . Needless to say, the fact that M is a free module implies severalsimplifications. Although it happens for the torus and the 3-sphere that their modulesof vector fields are free (i.e they are parallelizable manifolds), one expects a projectivemodule in general. However, as originally shown in the case of the noncommutative4-sphere [AW17a], real calculi can provide a way of performing local computations, inwhich case the (localized) module of vector fields is free. Definition 5.1.
A real calculus C A = ( A , g , M, ϕ ) is called free if there exists a basis ∂ , ..., ∂ m of g such that ϕ ( ∂ ) , ..., ϕ ( ∂ m ) is a basis of M as a (right) A -module.Note that if there exists a basis ∂ , . . . , ∂ m of g such that ϕ ( ∂ ) , . . . , ϕ ( ∂ m ) is a basisof M , then ϕ ( ∂ ′ ) , . . . , ϕ ( ∂ ′ m ) is a basis of M for any basis ∂ ′ , . . . , ∂ ′ m of g . Definition 5.2.
A real metric calculus ( C A , h ) is called free if C A is free and h isinvertible.An immediate consequence of having an invertible metric, is the existence of a Levi-Civita connection. Proposition 5.3.
Let ( C A , h ) be a free real metric calculus. Then there exists a uniqueaffine connection ∇ such that ( C A , h, ∇ ) is a pseudo-Riemannian calculus.Proof. Let { ∂ i } be a basis of g . Since C A is free it follows that E i = ϕ ( ∂ i ) provide abasis of M . In this basis one gets the components h ij = h ( E i , E j ) of the metric h , andfor notational convenience we set h i, [ j,k ] := h ( E i , ϕ [ ∂ j , ∂ k ]) and define K ijk ∈ A as K ijk := 12 (cid:0) ∂ i h jk + ∂ j h ik − ∂ k h ij − h i, [ j,k ] + h j, [ k,i ] + h k, [ i,j ] (cid:1) . Now, define the linear functional ˆ K ij ∈ M ∗ byˆ K ij ( E k b k ) := K ijk b k . Since the metric h is invertible, m ij = ˆ h − ( ˆ K ij ) ∈ M is well-defined, and2 h ( m ij , E k ) = 2ˆ h ( m ij )( E k ) = 2 ˆ K ij ( E k ) = 2 K ijk = ∂ i h jk + ∂ j h ik − ∂ k h ij − h i, [ j,k ] + h j, [ k,i ] + h k, [ i,j ] . From Corollary 2.13 it now follows that there exists a connection ∇ such that ( C A , h, ∇ )is pseudo-Riemannian, and from Theorem 2.10 it follows that ∇ is unique. (cid:3) Given a free real metric calculus ( C A , h ) and a basis ∂ , . . . , ∂ m of g , we write E a = ϕ ( ∂ a ) h ab = h ( E a , E b ) [ ∂ a , ∂ b ] = f cab ∂ c with f rpq ∈ R , giving h ( E a , ϕ ([ ∂ b , ∂ c ])) = h ar f rbc . The fact that ˆ h is invertible and { E a } ma =1 is a basis of M , implies that there exists h ab ∈ A such thatˆ h − ( ˆ E a ) = E b h ba ⇒ h ab = ˆ E a (cid:0) ˆ h − ( ˆ E b ) (cid:1) = h (cid:0) ˆ h − ( ˆ E a ) , ˆ h − ( ˆ E b ) (cid:1) where { ˆ E a } ma =1 is the basis of M ∗ dual to { E a } ma =1 . It follows that ( h ab ) ∗ = h ba and h ab h bc = h cb h ba = δ ac . For a free real metric calculus,we introduce the
Christoffel symbols Γ abc ∈ A as the(unique) coefficients ∇ b E c = E a Γ abc . Let us now derive an explicit formula for theChristoffel symbols in terms of the components of the metric. Indeed, by Koszul’sformula it follows that h ( E a Γ abc , E d ) = h ( ∇ b E c , E d ) = 12 ( ∂ b h cd + ∂ c h bd − ∂ d h bc − h br f rcd + h cr f rdb + h dr f rbc ) , and since the right hand side is hermitian, one obtains h da Γ abc = 12 ( ∂ b h cd + ∂ c h bd − ∂ d h bc − h br f rcd + h cr f rdb + h dr f rbc ) . Multiplying from the left by h pd givesΓ pbc = 12 h pd ( ∂ b h cd + ∂ c h bd − ∂ d h bc − h br f rcd + h cr f rdb ) + f pbc (5.1)and, in particular, if [ ∂ a , ∂ b ] = 0 for all a, b = 1 , . . . , m then(5.2) Γ abc = 12 h ad ( ∂ b h cd + ∂ c h bd − ∂ d h bc ) , in correspondence with the classical formula.Let ( C A , h ) and ( C A ′ , h ′ ) be free real metric calculi and let ( φ, ψ, b ψ ) : ( C A , h ) → ( C A ′ , h ′ ) be an isometric embedding. Since ψ is injective, it is easy to see that if { δ i } m ′ i =1 is a basis of g ′ , then { Ψ( δ i ) } m ′ i =1 is a basis of M Ψ , implying that M Ψ is a free moduleof rank m ′ . Let us now proceed to the define mean curvature, as well as minimality,of an embedding of free real metric calculi. Since we are working with extensions ofvector fields on the embedded manifold Σ ′ , rather than tangent vectors at points onΣ ′ , it is more natural to consider the restriction (to Σ ′ ) of the inner product of themean curvature vector with an arbitrary vector, rather than the mean curvature vectoritself. Definition 5.4.
Let ( C A , h ) and ( C A ′ , h ′ ) be free real metric calculi and let ( φ, ψ, b ψ ) :( C A , h ) → ( C A ′ , h ′ ) be an isometric embedding. Given a basis { δ i } m ′ i =1 of g ′ , the meancurvature H A ′ : M → A ′ of the embedding is defined as(5.3) H A ′ ( m ) = φ (cid:0) h (cid:0) m, α ( δ i , Ψ( δ j )) (cid:1)(cid:1) h ′ ij , giving trivially H A ′ ( m ) = 0 for m ∈ M Ψ . An embedding is called minimal if H A ′ ( ξ ) =0 for all ξ ∈ M ⊥ Ψ . Remark . Note that the ordering in (5.3) is natural in the following sense. Con-sidering the restriction of the metric h to M Ψ , given by h ij = h (Ψ( δ i )) , Ψ( δ j )) and itsinverse h ij , the fact that M is a right module gives a natural definition of the meancurvature as H A ′ ( m ) = φ (cid:0) h (cid:0) m, α ( δ i , Ψ( δ j )) h ij (cid:1)(cid:1) = φ (cid:0) h (cid:0) m, α ( δ i , Ψ( δ j )) (cid:1)(cid:1) φ ( h ij )= φ (cid:0) h (cid:0) m, α ( δ i , Ψ( δ j )) (cid:1)(cid:1) h ′ ij , reproducing the formula in Definition 5.4.Although defined with respect to a basis of g ′ , the mean curvature is independent ofthe choice of basis. Indeed, if we let h ′ ij and ˜ h ′ ij denote the components of the metric h ′ with respect to different bases { δ i } and { ˜ δ i } of g ′ , then there exists a (real) invertiblematrix A such that ˜ h ′ = Ah ′ A T , or equivalently ˜ h ′ ij = A ki h ′ kl A lj . Consequently,( ˜ h ′ ) ij = ( A − ) ik h ′ kl ( A − ) jl and it follows that the mean curvature calculated using the basis { ˜ δ i } is H A ′ ( m ) = φ (cid:16) h (cid:0) m, α (˜ δ i , Ψ(˜ δ j )) (cid:1)(cid:17) ( ˜ h ′ ) ij = φ (cid:0) h (cid:0) m, α ( A ki δ k , Ψ( A lj δ l )) (cid:1)(cid:1) ( A − ) im h ′ mn ( A − ) j n = A ki ( A − ) im A lj ( A − ) jn φ (cid:0) h (cid:0) m, α ( δ k , Ψ( δ l )) (cid:1)(cid:1) h ′ mn = φ (cid:0) h (cid:0) m, α ( δ k , Ψ( δ l )) (cid:1)(cid:1) ( h ′ ) kl showing that the definition of H A ′ is indeed basis independent.Let us end this section by noting that it is straight-forward to define the gradient,divergence and Laplace operator for free real metric calculi. Definition 5.6.
Let ( C A , h ) be a free real metric calculus and let ∇ denote the Levi-Civita connection. Moreover, let { ∂ a } ma =1 be a basis of g and set E a = ϕ ( ∂ a ). The gradient grad : A → M is defined asgrad( a ) = E a h ab ∂ b a for a ∈ A . The divergence div : M → A is defined asdiv( m ) = ( ∇ ∂ a m ) a for m ∈ M , where ∇ ∂ a m = E b ( ∇ ∂ a m ) b . The Laplace operator ∆ : A → A is definedas ∆( a ) = div (cid:0) grad( a ) (cid:1) for a ∈ A .Note that it is easy to check that the above definitions are independent of the choiceof basis of g . 6. Minimal tori in the 3-sphere
The 3-sphere has a rich flora of minimal surfaces, and the fact that minimal surfacesof arbitrary genus exist in S is a famous result by Lawson [Law70]. As an illustrationof the concepts we have developed, as well as being our motivating example, we shallconsider the noncommutative torus minimally embedded in the noncommutative 3-sphere. However, rather than the round metric on S , we will consider more generalmetrics. Therefore, let us start by recalling the classical situation. The Clifford torus T is embedded in S ⊆ R via ~x = ( x , x , x , x ) = (cos ϕ , sin ϕ , cos ϕ , sin ϕ ) . With δ = ∂ ϕ and δ = ∂ ϕ , the tangent space at a point is spanned by δ ~x = ( − sin ϕ , cos ϕ , ,
0) = ( − x , x , , δ ~x = (0 , , − sin ϕ , cos ϕ ) = (0 , , − x , x ) . The 3-sphere is embedded in C via z = e iξ sin ηw = e iξ cos η, and with ∂ = ∂ ξ and ∂ = ∂ ξ the tangent space at a point with 0 < ξ , ξ < π and0 < η < π/ E = ∂ ( x , x , x , x ) = ( − x , x , , E = ∂ ( x , x , x , x ) = (0 , , − x , x ) E η = ∂ η ( x , x , x , x ) = (cos ξ cos η, sin ξ cos η, − cos ξ sin η, − sin ξ sin η ) . The standard metric on S is given by g = sin η η
00 0 1 , and for H ∈ C ∞ ( S ) such that H > g = H sin η η
00 0 1 H. Let us now proceed to determine the Levi-Civita connection on ( S , ˜ g ). The Christoffelsymbols are computed usingΓ ijk = 12 ˜ g il ( ∂ j ˜ g kl + ∂ k ˜ g jl − ∂ l ˜ g jk ) , giving Γ jk = ∂ (ln H ) ∂ (ln H ) ∂ η (ln H ) + cot η∂ (ln H ) − ∂ (ln H ) cot η ∂ η (ln H ) + cot η − ∂ (ln H ) csc η Γ jk = − ∂ (ln H ) tan η ∂ (ln H ) 0 ∂ (ln H ) ∂ (ln H ) ∂ η (ln H ) − tan η ∂ η (ln H ) − tan η − ∂ (ln H ) sec η Γ jk = − ∂ η (ln H ) sin η − sin η cos η ∂ (ln H )0 − ∂ η (ln H ) cos η + sin η cos η ∂ (ln H ) ∂ (ln H ) ∂ (ln H ) ∂ η (ln H ) . Thus, the Levi-Civita connection is explicitly given as ∇ ∂ = ∂ (ln H ) ∂ − ∂ (ln H ) tan η∂ − ( ∂ η (ln H ) sin η + sin η cos η ) ∂ η ∇ ∂ = ∂ (ln H ) ∂ + ∂ (ln H ) ∂ = ∇ ξ ∂ ∇ ∂ η = ( ∂ η (ln H ) + cot η ) ∂ + ∂ (ln H ) ∂ η = ∇ η ∂ ∇ ∂ = − ∂ (ln H ) cot η∂ + ∂ (ln H ) ∂ + (sin η cos η − ∂ η (ln H ) cos η ) ∂ η ∇ ∂ η = ( ∂ η (ln H ) − tan η ) ∂ + ∂ (ln H ) ∂ η = ∇ η ∂ ∇ η ∂ η = − ∂ (ln H ) csc η∂ − ∂ (ln H ) sec η∂ + ∂ η (ln H ) ∂ η . Embedding the torus into the 3-sphere.
For fixed η ∈ (0 , π/ f η : T → ( S , ˜ g ) denote the embedding f η : (cos ϕ , sin ϕ , cos ϕ , sin ϕ ) → ( e iϕ sin η , e iϕ cos η ) , The induced metric on the torus is given by g T = ˜ H (cid:18) sin η
00 cos η (cid:19) ˜ H, where ˜ H ( ϕ , ϕ ) = H ( ϕ , ϕ , η ). The unit normal of T is N = ˜ H − ∂ η , and onewrites the second fundamental form α as: α ( δ , δ ) = − ˜ H ( ∂ η (ln H ) | η sin η + sin η cos η ) Nα ( δ , δ ) = α ( δ , δ ) = 0 α ( δ , δ ) = ˜ H (sin η cos η − ∂ η (ln H ) | η cos η ) N. Calculating the mean curvature of T in ( S , ˜ g ) yields H T = 12 ˜ g (cid:0) N, α (cid:0) δ i , δ j (cid:1)(cid:1) g ijT = − e − ˜ H (cot 2 η + ∂ η (ln H ) | η )and it follows that T is minimally embedded in ( S , ˜ g ) if ∂ η (ln H ) | η = − cot 2 η ; forinstance, one might choose H ( ξ , ξ , η ) = exp (cid:18) ˜ H ( ξ , ξ ) − r ( η ) cot 2 η r ′ ( η ) (cid:19) for an arbitrary function r with a nonzero derivative at η = η . In the classical case,when H = 1, the embedding is minimal if cot 2 η = 0, i.e. η = π/ The noncommutative minimal torus
Let us now apply the framework for noncommutative embeddings to the case of thenoncommutative torus and the noncommutative 3-sphere. We shall start by recallingtheir definitions, as well as their corresponding real metric calculi. For more details,we refer to [AW17b] (however, where only the standard metric on the 3-sphere wasconsidered). The noncommutative torus.
The noncommutative torus T θ is a unital ∗ -algebra generated by the unitary elements U, V subject to the relation
V U = qU V ,with q = e πiθ . Introducing the hermitian elements X = 12 ( U + U ∗ ) X = 12 i ( U − U ∗ ) X = 12 ( V + V ∗ ) X = 12 i ( V − V ∗ )gives = U U ∗ = ( X ) + ( X ) and = V V ∗ = ( X ) + ( X ) . In analogy with thegeometrical setting, let M ′ be the (right) submodule of ( T θ ) generated by e = ( − X , X , , e = (0 , , − X , X ) . We note that M ′ is a free T θ -module, since e and e form a basis for M ′ : e a + e b = 0 ⇒ ( − X a, X a, − X b, X b ) = (0 , , , ⇒ (cid:26) (cid:0) ( X ) + ( X ) (cid:1) a = U U ∗ a = a = 0 (cid:0) ( X ) + ( X ) (cid:1) b = V V ∗ b = b = 0 . Next, we let g ′ be the (real) Lie algebra generated by the two hermitian derivations δ , δ , given by δ U = iU δ V = 0 δ U = 0 δ V = iV, satisfying [ δ , δ ] = 0. Finally, let ϕ ′ : g ′ → M ′ with ϕ ′ ( δ j ) = e j for j = 1 , R -linearity, which implies that M ′ is generated by ϕ ′ ( g ′ ) as a T θ -module.Hence, we have shown that C T θ = ( T θ , g ′ , M ′ , ϕ ′ ) is a real calculus over the noncom-mutative torus.As a first illustration of a real calculus homomorphism, let us construct a family ofautomorphisms of T θ as follows. Let a, b, c, d ∈ Z be given such that ad − bc = 1, andlet α : T θ → T θ be the automorphism given by α ( U ) = U a V b ,α ( V ) = U c V d , with inverse α − ( U ) = q bd ( a − c − U d V − b α − ( V ) = q ac ( d − b − U − c V a . Once the automorphism α is established, it is a simple task to find a real calculusautomorphism from C T θ to itself by using Proposition 3.4 to find the required Liealgebra homomorphism. Indeed, Proposition 3.4 implies that ψ ( δ )( U ) = α ◦ δ ◦ α − ( U ) = idU ψ ( δ )( U ) = α ◦ δ ◦ α − ( U ) = − ibUψ ( δ )( V ) = α ◦ δ ◦ α − ( V ) = − icV ψ ( δ )( V ) = α ◦ δ ◦ α − ( V ) = iaV, giving ψ ( δ ) = dδ − cδ and ψ ( δ ) = − bδ + aδ . From the compatibility conditions b ψ (Ψ( δ i )) = ϕ ′ ( δ i ): b ψ (cid:0) e d − e c (cid:1) = e and b ψ (cid:0) − e b + e a (cid:1) = e one obtains b ψ ( e ) = e a + e c and b ψ ( e ) = e b + e d, ensuring that ( α, ψ, b ψ ) is an automorphism of the real calculus C T θ .7.2. The noncommutative 3-sphere.
The noncommutative 3-sphere S θ is the uni-tal ∗ -algebra generated by Z, Z ∗ , W, W ∗ satisfying W Z = qZW W ∗ Z = ¯ qZW ∗ W Z ∗ = ¯ qZ ∗ W W ∗ Z ∗ = qZ ∗ W ∗ Z ∗ Z = ZZ ∗ W ∗ W = W W ∗ W W ∗ = − ZZ ∗ , with q = e πiθ for θ ∈ R .Similar to the case of T θ , we introduce X = 12 ( Z + Z ∗ ) X = 12 i ( Z − Z ∗ ) X = 12 ( W + W ∗ ) X = 12 i ( W − W ∗ ) | Z | = ZZ ∗ | W | = W W ∗ , giving | Z | = ( X ) + ( X ) and | W | = ( X ) + ( X ) ; recall that | Z | and | W | arein the center of S θ and, furthermore, neither of them is a zero divisor (cf. [AW17b]).Let us now construct a real metric calculus for S θ , closely related to the Hopf fibrationof the 3-sphere.Recall from Section 6 that S can be given in terms of the coordinates ( ξ , ξ , η ),and we noted that the tangent plane at a given point is spanned by the three vectors E = ∂ ( x , x , x , x ) = ( − x , x , , E = ∂ ( x , x , x , x ) = (0 , , − x , x ) E η = ∂ η ( x , x , x , x ) = (cos ξ cos η, sin ξ cos η, − cos ξ sin η, − sin ξ sin η ) . For the noncommutative analogue, it is apparent how to choose E and E , but theanalogue of E η is less clear. Therefore, instead of ∂ η , one considers the derivation ∂ = | z || w | ∂ η , giving E = ∂ ( x , x , x , x ) = ( x | w | , x | w | , − x | z | , − x | z | ) , which can be used together with E and E to span the tangent space.Returning to the complex embedding coordinates z and w in C , one finds ∂ ( z ) = iz ∂ ( w ) = 0(7.1) ∂ ( z ) = 0 ∂ ( w ) = iw (7.2) ∂ ( z ) = z | w | ∂ ( w ) = − w | z | , (7.3) and with respect to the basis { E , E , E } of the tangent space of S , the inducedstandard metric becomes(7.4) ( h ab ) = ( h ( E a , E b )) = | z | | w |
00 0 | z | | w | . Motivated by the above considerations, let M the submodule of the free (right) module( S θ ) generated by { E , E , E } , where E = ( − X , X , , E = (0 , , − X , X ) E = ( X | W | , X | W | , − X | Z | , − X | Z | ) . In [AW17b] it was shown that M is a free module with a basis { E , E , E } and thatthere exist hermitian derivations ∂ , ∂ , ∂ such that ∂ ( Z ) = iZ ∂ ( W ) = 0 ∂ ( Z ) = 0 ∂ ( W ) = iW∂ ( Z ) = Z | W | ∂ ( W ) = − W | Z | , with [ ∂ a , ∂ b ] = 0 for a, b = 1 , ,
3. Let g be the (real) Lie algebra generated by ∂ , ∂ and ∂ , and define ϕ : g → M as the linear map (over R ) given by ϕ ( ∂ a ) = E a for a = 1 , ,
3. From the above considerations, it follows that C S θ = ( S θ , g , M, ϕ ) is a realcalculus over S θ .Now, let us proceed to construct a real metric calculus over S θ , in which we shallminimally embed the noncommutative torus. In analogy with Section 6, we choose thehermitian form h : M × M → Mh ( m, n ) = X a,b =1 ( m a ) ∗ h ab n b , where m = E a m a , n = E b n b and( h ab ) = H | Z | | W |
00 0 | Z | | W | H ∗ , where H ∈ S θ is chosen such that HH ∗ is invertible. Since neither | Z | nor | W | isa zero divisor, the metric is clearly nondegenerate; furthermore, h ab is hermitian for a, b = 1 , ,
3. We conclude that ( C S θ , h ) is a real metric calculus.Next, let us construct a metric and torsion-free connection on ( C S θ , h ). In orderachieve this, we will localize the algebra at | Z | and | W | . That is, one extendsthe algebra of the noncommutative 3-sphere by the inverses of | Z | and | W | . (Inprinciple, for a well-behaved noncommutative localization, one has to check the socalled Ore conditions, but since | Z | and | W | are central, these are trivially fulfilled.)The resulting algebra is denoted by S θ, loc . It is straight-forward to extend the realmetric calculus ( C S θ , h ) to a real metric calculus ( C S θ, loc , h ) (cf. [AW17a] where asimilar construction was carried out for the 4-sphere). Proposition 7.1.
There exists a unique affine connection ∇ such that ( C S θ, loc , h, ∇ ) is a pseudo-Riemannian calculus with ∇ E = E H − E | Z | | W | − H − E ( | W | − H + ) ∇ E = ∇ E = E H + E H ∇ E = ∇ E = E ( H + | W | ) + E H ∇ E = − E | W | | Z | − H + E H + E ( − | Z | − H ) ∇ E = ∇ E = E ( H − | Z | ) + E H ∇ E = − E | W | H − E | Z | H + E ( H + | W | − | Z | ) , where H a = ( HH ∗ ) − ∂ a ( HH ∗ ) for a = 1 , , .Proof. Since h is invertible, ( C S θ, loc , h ) is a free real metric calculus, implying that theLevi-Civita connection ∇ exists. Moreover, [ ∂ a , ∂ b ] = 0 for all a, b ∈ { , , } , and thusit follows that the Christoffel symbols for ∇ can be calculated directly using (5.2). Forinstance,Γ = 12 h ∂ h = 12 ( HH ∗ ) − ∂ ( HH ∗ ) = H Γ = 12 h ( − ∂ h ) = − | Z | | W | − ( HH ∗ ) − ∂ ( HH ∗ ) = −| Z | | W | − H Γ = 12 h ( − ∂ h ) = − | W | − ( HH ∗ ) − ∂ ( HH ∗ ) − = −| W | − H − , giving ∇ E = E H − E | Z | | W | − H − E ( | W | − H + ) . The remaining Christoffel symbols are computed in a completely analogous way. (cid:3)
An embedding of the noncommutative torus.
Finally, we will now constructan embedding ( φ, ψ, b ψ ) : C S θ, loc → C T θ . To this end, we set φ ( Z ) = λUφ ( W ) = µV, where λ and µ are complex nonzero constants such that | λ | + | µ | = 1. It is easy toverify that with these conditions φ is a ∗ -algebra homomorphism. Moreover, since λ and µ are chosen to be nonzero it means that φ is surjective as well. With this choiceof φ it follows that a Lie algebra homomorphism ψ : g ′ → g compatible with φ is givenby ψ ( δ ) = ∂ and ψ ( δ ) = ∂ , and M Ψ is the submodule of M generated by E and E . Furthermore, with b ψ ( E ) = e and b ψ ( E ) = e ( φ, ψ, b ψ ) is a real calculus homomorphism. This choice of ( φ, ψ, b ψ ) gives an embeddingof C T θ into C S θ, loc , since by choosing ˜ M to be the submodule of M generated by E one gets that M = M Ψ ⊕ ˜ M . Let us now find the induced metric h ′ such that ( φ, ψ, b ψ ) : ( C T θ , h ′ ) → ( C S θ, loc , h ) isan embedding of real metric calculi. Since M ′ has a basis { e , e } it suffices to calculate h ′ ( e i , e j ) for i, j = 1 , h ′ ( e , e ) = φ ( h ( E , E )) = φ (( HH ∗ ) | Z | ) = | λ | ( ˜ H ˜ H ∗ ) h ′ ( e , e ) = h ′ ( e , e ) = φ ( h ( E , E )) = 0 h ′ ( e , e ) = φ ( h ( E , E )) = φ (( HH ∗ ) | W | ) = | µ | ( ˜ H ˜ H ∗ ) , with ˜ H = φ ( H ); it is easy to check that h ′ is an invertible metric on M ′ , implying that( C T θ , h ′ ) is indeed a free real metric calculus. Moreover, it is clear that ˜ M = M ⊥ Ψ .Since M and M ′ are free modules, Proposition 4.3 can be used to quickly determinethe Levi-Civita connection ∇ ′ for ( C T θ , h ′ ): ∇ ′ e = b ψ ( L ( δ , Ψ( δ ))) = e ˜ H − e ˜ H | λ | | µ | − ∇ ′ e = ∇ ′ e = b ψ ( L ( δ , Ψ( δ ))) = e ˜ H + e ˜ H ∇ ′ e = b ψ ( L ( δ , Ψ( δ ))) = − e ˜ H | λ | − | µ | + e ˜ H , where ˜ H i = φ ( H i ) for i = 1 , ,
3. Consequently, one obtains the second fundamentalform as α ( δ , Ψ( δ )) = − E ( | W | − H + ) α ( δ , Ψ( δ )) = α ( δ , Ψ( δ )) = 0 α ( δ , Ψ( δ )) = E ( − | Z | − H ) , giving the mean curvature H T θ ( m ) = φ (cid:0) h (cid:0) m, α ( δ , Ψ( δ )) (cid:1)(cid:1) ( h ′ ) + φ (cid:0) h (cid:0) m, α ( δ , Ψ( δ )) (cid:1)(cid:1) ( h ′ ) = φ (cid:0) h (cid:0) m, − E ( | W | − H + ) (cid:1)(cid:1) | λ | − ( ˜ H ˜ H ∗ ) − + φ (cid:0) h (cid:0) m, E ( − | Z | − H ) (cid:1)(cid:1) | µ | − ( ˜ H ˜ H ∗ ) − = φ (cid:0) h (cid:0) m, E (cid:1)(cid:1) (cid:16) | µ | − − | λ | − − | λ | − | µ | − ˜ H (cid:17) ( ˜ H ˜ H ∗ ) − . For the embedded torus, M ⊥ Ψ is the submodule of M generated by the basis element E . Hence, the mean curvature is zero if0 = H T θ ( E ) = ( ˜ H ˜ H ∗ ) | λ | | µ | (cid:16) | µ | − − | λ | − − | λ | − | µ | − ˜ H (cid:17) ( ˜ H ˜ H ∗ ) − = ( ˜ H ˜ H ∗ ) (cid:16) | λ | − | µ | − H (cid:17) ( ˜ H ˜ H ∗ ) − = (cid:0) | λ | − | µ | (cid:1) −
2( ˜ H ˜ H ∗ ) ˜ H ( ˜ H ˜ H ∗ ) − = (cid:0) | λ | − | µ | (cid:1) − φ ( ∂ ( HH ∗ ))( ˜ H ˜ H ∗ ) − , implying that the embedding of ( C T θ , h ′ ) into ( C S θ , h ) is minimal if and only if φ (cid:0) ∂ ( HH ∗ ) (cid:1) = ( | λ | − | µ | ) φ ( HH ∗ ) . In the special case where φ ( ∂ ( HH ∗ )) = 0, the embedding is minimal if | λ | = | µ | =1 / √ | λ | and | µ | one may also choose, e.g., H = ZW giving HH ∗ = | Z | | W | and φ (cid:0) ∂ HH ∗ (cid:1) = 2 | λ | | µ | (cid:0) | µ | − | λ | (cid:1) = 0 . Acknowledgments
We would like to thank J. Choe for discussions. Furthermore, J.A. is supported by theSwedish Research Council grant 2017-03710.
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SIGMA , 9:071, 2013.(Joakim Arnlind)
Dept. of Math., Link¨oping University, 581 83 Link¨oping, Sweden
E-mail address : [email protected] (Axel Tiger Norkvist) Dept. of Math., Link¨oping University, 581 83 Link¨oping, Sweden
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