aa r X i v : . [ m a t h . G M ] S e p The heat flow in a parallelizable manifold
Erc¨ument Orta¸cgilSeptember 4, 2020
Abstract
We define the heat flow of a tensor field on a parallelizable manifoldand assign a precise Lie theoretic meaning to the process of ”smoothingout” as t → ∞ . Let f be a continuous real valued function defined on [0 , π ] , smooth on(0 , π ) and satisfying f ′′ ≡ f ( x ) = ax + b, x ∈ (0 , π ) . Nowsuppose we identify 0 and 2 π and assume that f is defined smoothly on thisidentification space. So we must have f (0) = f (2 π ) and therefore f ≡ b. Weconclude that if the second derivative of a smooth function on the unit circle S vanishes, then its first derivative vanishes too. In fact, we can replace f ′′ ≡ f ′′ ≥ f ′′ ≥ , then f is ”concaveup” on (0 , π ) and will form a ”sharp edge” at the point of identification unlessit is constant. Our first goal is to generalize this last statement from S toa compact manifold. The question is, of course, what we mean by first andsecond derivatives. To make progress with this question, we first recall thefundamental maximum/minimum principle which is the core of the so calledBochner technique. Proposition 1
Let U ⊂ R n be an open ball, g = ( g ij ( x )) a positive definitemetric on U and L a second order linear differential operator acting on smoothfunctions on U which is of the form L = g ab ( x ) ∂ ∂x a ∂x b + h a ( x ) ∂∂x a (1) for some smooth functions h i ( x ) on U, ≤ i ≤ n. Then, if L ( f ) ≥ on U and f attains a maximum in U (or L ( f ) ≤ on U and f attains a minimum in U ) , then f is constant and therefore L ( f ) = 0 on U. For the proof of Proposition 1, we refer to [Y], pg.26-29. We observe thatProposition 1 is a local assertion.Now we specialize to the case of a parallelizable manifold (
M, w ) where w is the structure object, i.e., a trivialization of the principal frame bundle of M. We recall that w defines a canonical metric g on M and a splitting ε : U → U . L def = g ab e ∇ a e ∇ b (2)where g = ( g ij ( x )) is the canonical metric defined by w ([O1], [O3]). We observethat L is a second order linear differential operator defined on global tensor fieldson ( M, w ) preserving the type of the tensor fields. In particular it is definedon functions on M. The key fact is that the restriction of L as an operator onfunctions to some coordinate neighborhood ( U, x ) ⊂ M is of the form (1). Thisfollows easily from (2), the definition of e ∇ ([O1], 5.8) and e ∇ i ( f ) = ∂f∂x i . In short, L is an elliptic operator. Definition 2 L is the e ∇ -Laplacian of the parallelizable manifold ( M, w ) . Atensor field α is e ∇ -harmonic if L ( α ) =0 . Clearly e ∇ i e ∇ j α = 0 implies L ( α ) =0 but not conversely. Therefore, e ∇ -harmonicity is a weaker condition than the vanishing of the ”second derivative”with the understanding that differentiation is given by e ∇ . Now we have the following fundamental fact.
Proposition 3
Let ( M, w ) be compact. If a function f satisfies L ( f ) ≥ (or L ( f ) ≤ on M, then f is constant on M and therefore L ( f ) = 0 . Proposition 3 follows immediately from the local fact stated by Proposition1 and the compactness of M. Indeed, suppose L ( f ) ≥ M and let A def = { x ∈ M | x is a local maximum of f } . Since M is compact, f attains a maximumsomewhere on M and therefore A = ∅ . It is easy to see that A ⊂ M is a closedsubset. We claim that A is also open. Indeed, if x ∈ A, we choose an open ball U around x and restrict L to U. Since L ( f ) ≥ U and L is of the form (1)on U as already observed above, f is constant on U by Proposition 1. It followsthat A is open and A = M since M is connected and therefore f is constant on M. For a tensor field α = ( α i ...i r j ...j s ( x )) of type ( r, s ) on ( M, w ) , its dual tensorfield α ⋆ = ( α j ...j s i ...i r ( x )) of type ( s, r ) is defined by lowering the upper indices andraising the lower indices of α as α ⋆ = α j ...j s i ...i r ( x ) def = g a i ...g a r i r g b j ...g b s j s α a ...a r b ...b s ( x ) (3)and its norm function k α k ( x ) is defined by k α k ( x ) def = α i ...i r j ...j s ( x ) α j ...j s i ...i r ( x ) def = α ( x ) · α ⋆ ( x ) (4)The main idea is now to apply Proposition 3 to the function f = k α k keepingin mind that e ∇ is a derivation on tensor algebra commuting with contractions.A straightforward computation now gives2 ∇ k e ∇ m k α k = (cid:16) e ∇ r e ∇ m α i ...i r j ...j s (cid:17) α j ...j s i ...i r + e ∇ m α i ...i r j ...j s e ∇ r α j ...j s i ...i r + e ∇ r α i ...i r j ...j s e ∇ m α j ...j s i ...i r + α i ...i r j ...j s e ∇ r e ∇ m α j ...j s i ...i r (5)Multiplying both sides of (5) with g km , summing over k, m, using e ∇ g = 0(recall that ∇ g = 0 in general where ∇ is defined by 5.26 in [O1]) and thenotation in (4), we deduce L ( k α k ) =2 L ( α ) · α ⋆ + 2 (cid:13)(cid:13)(cid:13) e ∇ α (cid:13)(cid:13)(cid:13) (6) Theorem 4
Let ( M, w ) be a compact parallelizable manifold and α a tensorfield on M. Then the following are equivalent.i) L ( α ) = 0 , i.e., α is e ∇ -harmonicii) e ∇ α = 0 , i.e., α is e ∇ -paralleliii) α is ε -invariant The equivalence of ii) and iii) is proved in [O1] (Proposition 5.5, pg.38)without the assumtion of compactness. Clearly ii) always implies i). To provei) ⇒ ii), (6) becomes now L ( k α k ) =2 (cid:13)(cid:13)(cid:13) e ∇ α (cid:13)(cid:13)(cid:13) ≥ f = k α k is constant on M by Proposition 3. It follows that L ( k α k ) = 0 which gives e ∇ α = 0 in view of (7). Corollary 5
Let ( M, w ) be a compact parallelizable manifold and α a tensorfield on M. If e ∇ ... e ∇ e ∇ ( α ) = 0 ( k -times, k ≥ ), then e ∇ ( α ) = 0 . Since e ∇ e ∇ ( α ) = 0 implies L ( α ) =0 , the conclusion for k = 2 follows from i) ⇒ ii) of Theorem 4 and the general case follows by an easy induction. In particular,let α = T = the torsion tensor of ( M, w ) (called the integrability object of w in [O2] and denoted by I ) . By Definition 6.1 in [O1], we have e ∇ T = R = thelinear curvature of ( M, w ) . At the end of Chapter 6 of [O1] we asked whether theassumption e ∇ e ∇ T = e ∇ R =0 gives an interesting theory other than the theoryof LLG’s. Corollary 5 gives a negative answer to this question for compact M. Now we come to a fundamental definition.
Definition 6
Let α be a tensor field on a parallelizable manifold ( M, w ) . Theheat flow of α is the second order linear PDE ∂α∂t = L ( α ( t )) α (0) = α (8)We will now state 3 heorem 7 On a compact ( M, w ) , the heat flow α ( t ) of α is uniquely definedfor all t ≥ and converges to a tensor field α ∞ as t → ∞ which satisfies L ( α ∞ ) = 0 , or equivalently, α ∞ is ε -invariant by iii) of Theorem 4. The proof of Theorem 7 follows the same lines as its Riemannian analog (seebelow) and contains no new ideas. Let T r,s denote the space of tensor fieldson M of type ( r, s ) and the subspace T r,sε ⊂ T r,s denote the ε -invariant tensorfields. Theorem 7 gives a linear map I : T r,s −→ T r,sε (9): α −→ α ∞ which fixes the subspace T r,sε since the heat flow (8) stabilizes this subspace, i.e.,this subspace is already the ”ideal limit” and is not touched by the heat flow. Itis standard to interpret heat-type equations as a ”rounding up” or ”smoothingout” or ”averaging” the initial condition as t → ∞ . As a remarkable fact, theseintuitive interpretations can be made very precise in the case of (8). For thispurpose, we now define another linear map I : T r,s −→ T r,sε as follows (see [O1],pg. 80-81): Let α ∈ T r,s , fix a base point e ∈ M and let T r,s ( e ) denote thespace of ( r, s )-tensors at e. We define the T r,s ( e )-valued function f on M by f ( x ) def = ε ( x, e ) ⋆ α ( x ) , i.e., f ( x ) ∈ T r,s ( e ) is the translation of the value α ( x )from x to e by the 1-arrow ε ( x, e ) from x to e. Now the integral R M f ( x ) dµ of f over M with respect to the unique normalized volume element µ defined bythe structure object w gives an element R M f ( x ) dµ def = α ∈ T r,s ( e ) which isthe total average of α over M. We now distribute α over M using ε, i.e., wedefine the ε -invariant tensor field α ,ε on M by α ,ε ( x ) def = ε ( e, x ) ⋆ α , x ∈ M, which is easily seen to be independent of the base point e. This gives the linearmap I : T r,s −→ T r,sε (10): α −→ α ,ε which again fixes T r,sε . We observe that this averaging process does not needa global compact Lie group but needs only the compact parallelizable (