CR extensions with a classical Several Complex Variables point of view. August Peter Brådalen Sonne Master s Thesis, Spring 2018

Transcription

1 CR extensions with a classical Several Comlex Variables oint of view August Peter Brådalen Sonne Master s Thesis, Sring 2018

2 This master s thesis is submitted under the master s rogramme Mathematics, with rogramme otion Mathematics, at the Deartment of Mathematics, University of Oslo. The scoe of the thesis is 60 credits. The front age deicts a section of the root system of the excetional Lie grou E 8, rojected into the lane. Lie grous were invented by the Norwegian mathematician Sohus Lie ( ) to exress symmetries in differential equations and today they lay a central role in various arts of mathematics.

3 Acknowledgements I would like to give a secial thanks to Berit Stensønes for her eternal atience and for always ushing me in the right direction. 1

4 Contents 1 Introduction 3 2 Holomorhic functions and domains of holomorhy Holomorhic functions in several comlex variables Domains of holomorhy Enveloes of holomorhy Embedded submanifolds of C n Real embedded submanifolds of C n The comlex tangent sace CR submanifolds Comlex submanifolds of C n The Levi form for generic CR-submanifolds Involutive and integrable subbundles The Levi form for C 2 -smooth generic CR-submanifolds CR functions and the Baouendi-Treves aroximation theorem 50 6 The construction of analytic disks attached to a generic CRsubmanifold The Hilbert transform and Privalov s theorem Bisho s equation and the construction of analytic disks An indication of the roof of Tréreau s theorem for hyersurfaces in C

5 1 Introduction This thesis is dedicated to develoing the theory required in order to rove a theorem first roven by Tréreau in [13], stating that if M C n is a real hyersurface. Then all function f : M C satisfying the tangential Cauchy-Riemann equations extend holomorhically to one side of M, if and only if M is not seudoconvex from each side. In chater 1 we start off by recalling some basic roerties of holomorhic functions in several comlex variables as well as the basic theory on domains of holomorhy following the resentations in [5] and [11]. We then introduce some of the basic theory of enveloes of holomorhy of schlicht domains. Insired by an examle in [4] of a domain which has no enveloe of holomorhy we introduce the Hartogs hull oerator which leads to a useful tool for showing that certain domains are schlicht. In chater 2 we study real embedded submanifolds in C n loosely following a book the resentation in [1]. We start off by introducing the comlex tangent saces of a real embedded submanifold for which we offer several alternate descritions. Insired by an examle of a submanifold for which the dimension of the comlex tangent sace deends on the oint we then introduce the concet of an embedded CR-submanifold. The embedded CR-submanifolds may be thought of as the class of embedded submanifolds for which we may define Cauchy-Riemann equations. Imortant examles of embedded CR-submanifolds include the real hyersurfaces as well as the comlex submanifolds. Finally we define the concet of a generic CR-submanifold and show that all such submanifolds may be brought on a articularly simle form by a local biholomorhic change of variables. In chater 3 we introduce the Lie bracket for vector fields over C 2 -smooth embedded submanifolds of C n which in combination with Frobenius theorem gives a necessary and sufficient condition for the existence of a submanifold with a rescribed tangent bundle structure. We then introduce the concet of a Levi-flat CR-submanifold and we show that the Levi-flatness of a generic CR-submanifold is equivalent to the vanishing of a generalized Levi form. In chater 4 we introduce the concet of a CR-function which generalizes the concet of a holomorhic function. Exanding uon the roof in [1] we establish the Baouendi-Treves aroximation theorem for embedded CR-submanifolds. This result states that if M C n is a C 2 -smooth generic CR-submanifold, then each oint M is contained in an oen neighborhood ω M where all CR-functions may be uniformly aroximated by a sequence of entire holomorhic functions. We then introduce the concet of an analytic disk attached to M and we show that the aroximating functions from Baouendi-Treves aroximation theorem in fact converge to be holomorhic on the (ossibly emty) interior of all such analytic disks locally attached to M. 3

6 In chater 5 we study an imortant aer by Hill-Taiani [7] on the existence of analytic disks attached to a given C 2 -smooth generic CRsubmanifold. We start off by recalling some elementary roerties of harmonic functions. We then introduce the Hilbert transform on the unit circle which we use in order to formulate the Bisho equation. This functional equation gives a necessary and sufficient condition for the existence of (sufficiently regular) analytic disks attached to a generic C 2 -smooth generic CR-submanifold. By a modification of the roof given in [7] of Privalov s theorem we show that the Hilbert transform is a linear continuous oerator between Banach saces. In combination with the imlicit function theorem for Banach saces this leads to the imortant existence theorem for solutions to the Bisho equation. We then introduce a result by Hill-Taiani which shows how the existence of solutions to the Bisho equation may be used in order to find a simle sufficient condition for the existence of a family of analytic disks attached to a given C 5 -smooth generic CR-submanifold, so that the disks foliate a higher-dimensional submanifold. Finally we mention how this result may be used in order to rove a weak version of Tréreau s theorem, and we suggest how one may roceed in order to give a roof of Tréreau s theorem for hyersurfaces in C 2. 2 Holomorhic functions and domains of holomorhy In this chater we recall some basic roerties holomorhic functions and domains of holomorhy. The results and roofs of the results in the first two sections are well-known and may be found in [5] or [11]. In section three we introduce the concet of the enveloe of holomorhy for a connected domain Ω C n as the maximal domain Ω Ω where all holomorhic functions f : Ω C extend holomorhically. Insired by an examle in [4] of a domain which has no enveloe of holomorhy we introduce the concet of a schlicht domain. Finally we introduce the Hartogs hull oerator which gives a way of constructing the enveloe of holomorhy of a schlicht domain, as well as a way of showing that certains domains are schlicht. 2.1 Holomorhic functions in several comlex variables Definition 1. We define the Wirtinger artial derivatives in C n as the oerators defined for 1 j n by = 1 ( i ), = 1 ( + i ). (1) z j 2 x j y j z j 2 x j y j Definition 2. If f : Ω C is a differentiable function defined in an oen 4

7 set Ω C n, then we define the differential of f to be the one-form n f df = dx j + f dy j. (2) x j y j j=1 Introducing the comlex one-forms dz j = dx j + idy j and d z j we obtain the following. Theorem 2.1. Let f : Ω C be a differentiable function defined in an oen set Ω C n, then df = f + f, where n f n f f = dz j, f = d z j. (3) z j z j j=1 We may regard the one-forms dz j and dz j as mas dz j : C n C and dz j : C n by letting dz j (z 1,..., z n ) = z j and letting dz j (z 1,..., z n ) = z j. In this case it is easy to show that the ma f : C n C is C-linear, while the ma f : C n C is antilinear. Definition 3. Let f : Ω C be a continuous function defined on an oen set Ω C n. We say that the function is holomorhic if f = 0, or equivalently if we for each 1 j n and each Ω have that j=1 f z j () = 0. (4) We shall refer to this last system of differential equations as the Cauchy- Riemann equations. It should be noted that the condition that f : Ω C is continuous is not required. This follows from Hartog s theorem on searate analyticity which states that a function f : Ω C is holomorhic if and only if it is holomorhic in each variable searately. It is easy to see that the collection of holomorhic functions on a domain has a natural ring structure. Definition 4. Let Ω C n be a domain, then we define O(Ω) to be the ring of holomorhic functions f : Ω C. Definition 5. We define the olydisk centered at C n of olyradius r R n + as the set n (, r) = {z C n z j j < r j for 1 j n}. (5) In the secial case where = 0 and where r = 1 we shall refer to n (0, 1) as the unit olydisk in C n. In addition we define the distinguished boundary of the olydisk n (, r) as the set Γ n (, r) = {z C n z j j = r j for all 1 j n}. (6) 5

8 In the case where n = 1 we see that the unit olydisk 1 (0, 1) C is simly the oen unit disk. We shall denote the oen unit disk by D and we shall denote its boundary by S 1. By alying the single-variable Cauchy integral formula in each variable searately one easily obtains the following. Theorem 2.2. (The Cauchy integral formula) Let f : Ω C be a holomorhic function defined in an oen set Ω C n and let n (, r) Ω be a relatively comact olydisk. If z n (, r) then f(z) = ( ) 1 n 2πi ζ Γ n (,r) f(ζ) dζ. (7) (ζ n z 1 ) (ζ n z n ) By differentiating the above exression one easily shows that any holomorhic function is C -smooth. Conversely if n (, r) is a olydisk and if f : n (, r) C is a continuous function.then the Cauchy-Riemann equations that the function F : n (, r) C given by F (z) = ( ) 1 n 2πi ζ Γ n (,r) is holomorhic. This easily imlies the following. f(ζ) dζ, (8) (ζ 1 z 1 ) (ζ n z n ) Theorem 2.3. Let Ω C n be an oen set and let {f j } j N be a sequence of holomorhic functions f j : Ω C. If the functions converge uniformly on comacts to a function f : Ω C, then f is holomorhic. By a similar argument as in the one-variable case involving the Cauchy integral formula (Theorem 2.2) one easily shows the following. Theorem 2.4. Let f : Ω C be a holomorhic function defined in an oen set Ω C n. If n (, r) Ω is a olydisk relatively comact in Ω, then the ower series F (z) = α N n 0 converges uniformly to f in n (, r). D α f() α 1! α n! (z 1 1 ) α1 (z n n ) αn, (9) The ower series exansion of holomorhic functions easily imlies the identity theorem. Theorem 2.5. (The identity theorem) Let f : Ω C be a holomorhic function defined in an oen connected set Ω C n. If f vanishes on an oen set U Ω, then f is identically zero. 6

9 The concet of holomorhicity extends readily to vector-valued functions. Definition 6. Let f : Ω C m be a continuous function defined in an oen set Ω C n. We say that f is holomorhic if for each 1 j m we have that the function f j : Ω C is holomorhic. It is convenient to introduce the holomorhic equivalent of a diffeomorhism. Definition 7. Let f : Ω C n be a holomorhic function defined in an oen set Ω C n. If f is invertible with a holomorhic inverse, then we say that f is biholomorhic. 2.2 Domains of holomorhy For any bounded domain Ω C we may find a sequence of oints {z j } j N Ω which clusters at each boundary oint. By the Weierstrass roduct theorem this suggests the existence of a holomorhic function f : Ω C which tends to zero on the boundary. Since holomorhic functions in C have isolated zeroes this imlies that f does not extend to be holomorhic ast the boundary. This suggests that there exist no strictly larger domain Ω Ω so that we may identify O(Ω) = O( Ω). As we shall see the situation is radically different in higher dimensions. There exist bounded domains Ω Ω C n for each n 2 so that O(Ω) = O( Ω). If Ω C n is a bounded domain, then a sufficient condition for the nonexistence of a domain Ω Ω so that O(Ω) = O( Ω) is the existence of a holomorhic function f : Ω C for each Ω so that the function f does not extend to be holomorhic ast the oint. This motivates the following definition. Definition 8. Let Ω C n be an oen set and let f : Ω C be a holomorhic function. We say that f is comletely singular at a oint Ω if there exists no holomorhic function f : U C defined in an oen connected neighborhood U C n of which agrees with f on U Ω. Note that if f : Ω C is not comletely singular at a oint M, then there exists an oen connected neighborhood U C n of and a holomorhic function f : U C which agrees with f on the oen connected set Ω U. By the identity theorem for holomorhic functions this imlies the existence of a local holomorhic extension of f ast the oint. Definition 9. Let Ω C n be an oen set, we say that Ω is a weak domain of holomorhy if there for each oint Ω exists a holomorhic function f : Ω C which is comletely singular at. 7

10 By an earlier remark we see that a domain Ω C n is a weak domain of holomorhy if there for each boundary oint Ω exists a holomorhic function f : Ω C which does not extend to be holomorhic ast the oint. If Ω C n is a convex oen set and if Ω. real-valued linear function f : C n R of the form Then there exists a f(x 1 + iy 1,..., x n + iy n ) = n a j x j + b j y j, (10) so that f() = 0 and so that f(z) < 0 for all z Ω. This may be used in order to rove the following. Theorem 2.6. Let Ω C n be a convex oen set. Then Ω is a weak domain of holomorhy. Definition 10. Let Ω C n be an oen set, we say that Ω is a domain of holomorhy if there exists a holomorhic function f : Ω C which is comletely singular at each boundary oint Ω. It is clear that any domain of holomorhy is a weak domain of holomorhy. In fact one can show that the converse also holds; any weak domain of holomorhy is necessarily a domain of holomorhy. Definition 11. Let Ω C n be an oen set, we say that Ω is a local domain of holomorhy if each oint Ω is contained in an oen neighborhood U C n so that Ω U is a domain of holomorhy. Definition 12. We define a Euclidean Hartogs figure to be a air ( n (0, 1), H) where H n (0, 1) is a set of the form H = {z n (0, 1) r n < z n } {z n (0, 1) z j < r j for 1 j n 1}. (11) For some ositive real numbers r 1,... r n satisfying 0 < r j < 1. Theorem 2.7. Let n 2, let ( n (0, 1), H) be a Euclidean Hartogs figure and let f : H C be a holomorhic function. Then there exists a holomorhic function f : n (0, 1) C which extends f. Proof. Since ( n (0, 1), H) is a Euclidean Hartogs figure there exist real numbers r 1,..., r n satisfying 0 < r j < 1 so that H = {z n (0, 1) r n < z n or z j < r j for 1 j n 1}. (12) j=1 8

11 Define ɛ = min 1 j n r j and ick some real number so that r n < r < 1. Now let D r = {z n (0, 1) r n < r}. Define the function f : D r C by letting f (z) = 1 f(z 1,..., z n 1, ζ) dζ. (13) 2πi ζ =r ζ z n Differentiating under the integral we easily verify that f is holomorhic. Moreover by the single-variable Cauchy-Integral formula it is clear that f (z) = f(z) for all oints z (0, ɛ). It follows by the identity rincile that f and f agree on their common domain of definition. This imlies that the function { f f(z) (z) for z D r = (14) f(z) for z H defines a holomorhic extension of f to D r H = n (0, 1). The above theorem suggests that if ( n (0, 1), H) is a Euclidean Hartogs figure with n 2, then H is not a domain of holomorhy. Definition 13. Let ( n (0, 1), H) be a Euclidean Hartogs figure and let F : n (0, 1) F ( n (0, 1)) be a biholomorhism. Then we refer to the air ( n, H) = (F ( n (0, 1)), F (H)) as a general Hartogs figure. The extension theorem for Euclidean Hartogs figures extends easily to general Hartogs figures. Theorem 2.8. Let n 2, let ( n, H) be a general Hartogs figure and let f : H C be a holomorhic function. Then there exists a holomorhic function f : n C which extends f. Proof. Since ( n, H) is a general Hartogs figure there exists a Euclidean Hartogs figure ( n (0, 1), H) and a biholomorhic function F : n (0, 1) C n so that F (H) = H and so that F ( n (0, 1)) = n. It follows that the function G = (f F ) H : H C is a holomorhic function which by Theorem 2.7 has a holomorhic extension G: n (0, 1) C. Now consider the holomorhic function f = G F 1 : n C, then f H = (G F 1 ) H = f. It follows that f is a holomorhic extension of f. Definition 14. Let Ω C n be an oen set. We say that Ω is Hartogsseudoconvex if for each general Hartogs figure ( n, H) with H Ω we have that n Ω. It is easy to see that a domain of holomorhy Ω C n is necessarily Hartogs-seudoconvex. If this were not the case then there would exist a general Hartogs figure ( n, H) so that H Ω while n Ω. Letting Ω H and alying Theorem 2.7 this would imly that any holomorhic function f : Ω C would extend ast. 9

12 We shall soon see that domains of holomorhy may also be characterized by a form of disk convexity. We first introduce the concet of an analytic disk. Definition 15. If ψ : D C n is a continuous ma which is holomorhic on D, then we refer to ψ as an analytic disk. We shall refer to the ma ψ S 1 as the boundary of the analytic disk. We shall frequently identify an analytic disk ψ : D C n with its image ψ ( D ) C n. Similarly we shall oftentimes refer to the image ψ(s 1 ) as the boundary of the analytic disk. Definition 16. Let ψ : [0, 1] D C n be a continuous function so that for each t [0, 1] the ma ψ t : z ψ(t, z) is an analytic disk. Then ψ is called a continuous family of analytic disks. Definition 17. Let Ω C n be an oen set. We say that Ω has the disk roerty if for each continuous family of analytic disks ψ : [0, 1] D C n with ψ ( {0} D ) ψ ( [0, 1] S 1) Ω (15) it follows that ψ ( [0, 1] D ) Ω. (16) It should be mentioned that there exist several (equivalent) definitions of the disk roerty. The above definition is taken from [5]. Definition 18. Let Ω C n be an oen set and let f : Ω R { } be a function. We say that f is uer semicontinuous if for each c R the set {z Ω f(z) < c} is oen. It is obvious that any continuous function is necessarily uer semicontinuous. In addition one can show that if f : Ω R { } is uer semicontinuous then for each comact K Ω we have that f dλ <. (17) K Definition 19. Let Ω C be an oen set. An uer semicontinuous function f : Ω R { } is called subharmonic if for each oint z Ω there exists some ρ > 0 so that for all 0 < r ρ one has f(z) 1 2π f(z + re iθ ) dθ. (18) 2π 0 It is not hard to show that subharmonic functions satisfy the maximum rincile. 10

13 Theorem 2.9. Let Ω C n be a connected oen set and let f : Ω R { } be a subharmonic function. If w Ω is a oint so that f(w) f(z) for all z Ω, then f is constant. The following result gives a simle way of classifying sufficiently regular subharmonic functions. Theorem Let Ω C n be an oen set and let f : Ω R be a C 2 - smooth function. The function f is subharmonic if and only if its Lalacian satisfies f 0. The natural extension of subharmonicity to several variables is lurisubharmonicity. Definition 20. Let Ω C n be an oen set. An uer semicontinuous function f : Ω R { } is called lurisubharmonic if for each z Ω and for each w C n one has that the function ζ f(z + ζw) is subharmonic wherever it is defined. We are now ready to define the concet of seudoconvexity. Definition 21. Let Ω C n be an oen set and let d Ω : Ω R { } be the distance function d Ω (z) = inf w Ω z w C n. (19) We say that Ω is seudoconvex if the function log d Ω : Ω R { } is lurisubharmonic. It should be noted that the seudoconvexity of a domain Ω C n is tyically defined by the existence of a lurisubharmonic exhaustion function. These two definitions can however be shown to be entirely equivalent. It can be shown that if f, g : Ω R { } are lurisubharmonic functions, then so is max {f, g} : Ω R { }. This imlies the following. Theorem Let Ω 1 C n and Ω 2 C n be seudoconvex domains. Then Ω 1 Ω 2 is seudoconvex. The famed solution to the Levi roblem gives us several ways of verifying that a domain is a domain of holomorhy. Theorem (The solution to the Levi roblem) Let Ω C n be an oen set, then the following are equivalent. (i) Ω is a weak domain of holomorhy. (ii) Ω is a domain of holomorhy. 11

14 (iii) Ω is a local domain of holomorhy. (iv) Ω is Hartogs seudoconvex. (v) Ω has the disk roerty. (vi) Ω is seudoconvex. For oen sets Ω C n with C 2 -smooth boundary we may define domains of holomorhy by a simle differential criterion. If Ω, then there exists a C 2 -smooth function r : U R defined in an oen neighborhood U C n of so that (i) Ω U = {z U r(z) = 0}, (ii) Ω U = {z U r(z) < 0}, (iii) dr 0 in U. We refer to any such function as an oriented locally defined defining function for Ω at. Furthermore we define the holomorhic tangent sace at as the subsace T (1,0) Ω C n given by n T (1,0) Ω = a r Cn a j () = 0 z j. (20) It can be shown that this subsace is a comlex (n 1)-dimensional subsace which is indeendent of the function r. Definition 22. Let Ω C n be an oen set with C 2 -smooth boundary. We say that Ω is Levi-seudoconvex if there for each oint Ω exists an oriented locally defined defining function r : U R for Ω at so that the Levi form L r : T (1,0) Ω R given by is non-negative. L r(a 1,... a n ) = j=1 n j=1 k=1 2 r z j z k ()a j a j, (21) It turns out that the condition that L r 0 is indeendent of the choice of function r. In fact the following holds. Theorem Let Ω C n be an oen set with C 2 -smooth boundary, then Ω is seudoconvex if and only if it is Levi-seudoconvex. Note that by the solution to the Levi roblem this gives a simle way of verifying whether a C 2 -smooth domain is a domain of holomorhy. 12

15 2.3 Enveloes of holomorhy In Theorem 2.7 we saw that if ( n (0, 1), H) is a Euclidean figure with n 2, then all holomorhic functions f : H C admit a holomorhic extension f : n (0, 1) C. This suggests the identification O(H) = O( n (0, 1)). Furthermore, since n (0, 1) is convex it follows from Theorem 2.6 that it is seudoconvex. This imlies that there exist no strictly larger domains Ω n (0, 1) with O( Ω) = O(H). In this sense we see that n (0, 1) is the maximal extension domain of H. We make the following definition. Definition 23. Let Ω C n be a connected domain. Suose that there exists a seudoconvex connected set Ω Ω so that O(Ω) = O( Ω). Then we refer to Ω as the enveloe of holomorhy of Ω. Since the enveloe of holomorhy of a domain is suosed to reresent the maximal extension domain we should require that it is unique. The following theorem shows that this is indeed the case. Theorem Let Ω C n be a connected domain with enveloes of holomorhy Ω 1 and Ω 2, then Ω 1 = Ω 2. Proof. If Ω C n is a connected domain with enveloes of holomorhy Ω 1 and Ω 2. Then we may for each holomorhic function f : Ω C find holomorhic functions f 1 : Ω 1 C and f 2 : Ω 2 C which agree with f on Ω. Since Ω 1 and Ω 2 are connected sets containing Ω we see that Ω 1 Ω 2 is connected. It follows by an alication of the identity theorem (Theorem 2.5) that the function f : Ω 1 Ω 2 C given by f(z) = { f1 (z) for z Ω 1 f 2 (z) for z Ω 2 (22) is a well-defined holomorhic function. This construction suggests the identification O( Ω 1 ) = O( Ω 2 ) = O( Ω 1 Ω 2 ). Now if Ω 1 Ω 2, then there exists (after a ossible relabeling) some oint Ω 1 Ω 2. The above construction suggests that all holomorhic functions on Ω 1 extend to be holomorhic on the connected set Ω 1 Ω 2 which imlies that the domain Ω 1 is not a weak domain of holomorhy. By the solution to the Levi roblem (Theorem 2.12) this contradicts the seudoconvexity of Ω 1. We must therefore conclude that Ω 1 = Ω 2. It is imortant to note that there exist connected domains Ω C n which admit no enveloe of holomorhy, an examle of such a domain may be found in [4]. This can be remedied if we allow our holomorhic functions to extend holomorhically to a secial kind of n-dimensional comlex manifold. We shall not need to do so, instead we introduce the following definition. 13

16 Definition 24. Let Ω C n be a connected domain whose enveloe of holomorhy exists. Then we refer to Ω as a schlicht domain. We will need the following roerty of schlicht domains. Theorem Let Ω C n be a schlicht domain with enveloe of holomorhy Ω C n. If f : Ω C is a holomorhic function, then f(ω) = f( Ω). Proof. If this is not the case then there exists some oint w Ω \ Ω so that f(w) / f(ω). Now define the holomorhic function g : Ω C by letting g(z) = 1 f(z) f(w). (23) By the definition of the enveloe of holomorhy this function has a holomorhic extension g : Ω C. Alying the identity theorem we see that f(z) = 1 + f(w). (24) g(z) Evaluating this exression at z = w gives that 1/ g(w) = 0 which is clearly absurd. We must therefore conclude that f(ω) = f( Ω). In general it is not a simle task to verify whether that a domain is schlicht. In order to give one way of showing that a domain is schlicht we introduce the Hartogs hull oerator which is insired by the fact that a domain is seudoconvex if and only if it is Hartogs seudoconvex. Definition 25. Let D n denote the collection of connected sets in C n. We define the Hartogs hull oerator T : D n D n by letting T (Ω) be the collection of all oints z C n that are contained in some n, where ( n, H) is a general Hartogs figure with H Ω. It is easy to see that the Hartogs hull oerator is well-defined. If Ω C n is a connected domain then we define a sequence of nested oen connected sets Ω = T 0 (Ω) T (Ω) T 2 (Ω)... T j (Ω)... (25) by the recursive relation T k+1 (Ω) = T (T k (Ω)). The imortance of the Hartogs hull oerator follows from the following theorem. Theorem Let Ω C n be a connected domain and let T (Ω) = j N T j (Ω). (26) Then T (Ω) is a seudoconvex connected domain. Moreover if Ω is schlicht, then T (Ω) is its enveloe of holomorhy. 14

17 Proof. We first show that if Ω C n is any domain, then T (Ω) is a seudoconvex connected set. The fact that T (Ω) is a connected domain follows easily from a toological argument. To see that it is also seudoconvex we recall from the solution to the Levi roblem (Theorem 2.12) that T (Ω) is seudoconvex if and only if it is Hartogs seudoconvex. Now if ( n, H) is a general Hartogs figure with H T (Ω), then H T k (Ω) for some k N. It follows from the definition of the Hartogs hull oerator that n T k+1 (Ω) T (Ω). Now suose that Ω is a schlicht domain with enveloe of holomorhy Ω, we need to show that Ω = T (Ω). By another alication of the solution to the Levi roblem we see that the enveloe of holomorhy Ω is Hartogs seudoconvex. Since Ω Ω this imlies that T j (Ω) Ω for all j N which shows that T (Ω) Ω. Suose now that this inclusion is strict, then by another alication of the solution to the Levi roblem we see that the set T (Ω) is a domain of holomorhy contained in Ω. It follows that there exists a holomorhic function f : T (Ω) C which does not extend to be holomorhic on any Ω. But then the function f Ω : Ω C is a holomorhic function which has no holomorhic extension to the enveloe of holomorhy Ω which contradicts the assumtion that Ω is the enveloe of holomorhy of Ω. We must therefore conclude that T (Ω) = Ω. It is interesting to note that even for a non-schlicht domain Ω C n the set T (Ω) is always seudoconvex. The reason for T (Ω) not being the enveloe of holomorhy of Ω must therefore stem from the fact that there exists some k N 0 for which not all holomorhic functions f : T k (Ω) C extend to be holomorhic on T k+1 (Ω). The existence of non-schlicht domains might therefore aear to contradict Theorem 2.8 which states that if ( n, H) is a general Hartogs figure with H T k (Ω). Then each holomorhic function f : H C has a holomorhic extension f : n C. This gives a local holomorhic extension of f to an oen set in T k+1 (Ω), but it may fail to give a global holomorhic extension to all of T k+1 (Ω). To examine what may go wrong we attemt to define a global holomorhic extension f : T k+1 (Ω) C in the following way. If z T k+1 (Ω) then there exists a general Hartogs figure ( n z, H z ) so that H z T k (Ω) and so that z n z. Now consider the holomorhic function f Hz : Hz C and let f z : n z C be the holomorhic extension from Theorem 2.8. We wish to define a global holomorhic extension of f by defining f(z) = fz (z), where { fz f(z) for z Ω, (z) = f z (z) for z n (27) z. 15

18 If the functions fz are well-defined then by a connectedness argument and the identity rincile one sees that this indeed gives a well-defined holomorhic extension. This need however not be the case, in order to show that the functions fz are well-defined we would have to show that for all z T k (Ω) n z we have that f(z) = f z (z). If the set T k (Ω) n z is connected then this follows from the identity rincile, but in the case where the set fails to be connected the function need not be well-defined. This is exactly what goes wrong in the examle of a non-schlicht domain Ω C 2 given in [4] where the function f(z, w) = z 3 has a well-defined local holomorhic extension at each oint but it fails to form a global holomorhic extension. The above idea immediately gives the following interesting result. Theorem Let D D be a class of connected oen domains with the following roerties: (i) If Ω D, then T (Ω) D. (ii) For each Ω D and each general Hartogs figure ( n, H) with H Ω, we have that Ω n is connected. Then each element Ω D is schlicht. Proof. Let Ω D, then Ω is schlicht if and only if we for each k N 0 and each holomorhic function f : T k (Ω) C have a holomorhic extension f : T k+1 (Ω) C. By the above discussion this will be the case if we can show that for any general Hartogs figure ( n, H) with H T k (Ω) we have that T k (Ω) n is a connected set. This follows immediately from the second roerty of the class D. In fact by the solution to the Levi roblem we have that a domain is seudoconvex if and only if it is seudoconvex locally. This can be used in order to define a generalized Hartogs hull oerators T ɛ : D n D n for 0 < ɛ by only considering general Hartogs figures whose diameter is smaller than ɛ. It is not hard to show that these oerators also lead to a way of constructing the enveloe of holomorhy of schlicht domains. 3 Embedded submanifolds of C n We shall assume that the reader is familiar with the concet of real and comlex differentiable manifolds (both with and without boundary), differentiable functions between such manifolds, as well as some basic knowledge of real and comlex differential forms. In this chater we introduce the concet of an embedded submanifold of C n and we study its real and comlexified tangent sace. We then introduce a secial class of embedded submanifolds known as CR-submanifolds whose geometric tangent bundle 16

19 contains a comlex subbundle. Imortant examles of CR-submanifolds include the real hyersurfaces and comlex submanifolds. Many of the results in this chater are taken from [1], but we offer several anternate, missing and exanded roofs. 3.1 Real embedded submanifolds of C n Definition 26. Let M C n, we say that M is a C s -smooth (1 s ) embedded submanifold of C n of codimension 0 d 2n if for each M there exists an oen set U C n, and a C s -smooth function r : U R d which satisfies the following: (i) M U = {z U r(z) = 0}. (ii) dr 1 dr d 0 on U. We refer to any such function r as a locally defined defining function for M at. It can be shown that if M C n is a C s -smooth (1 s ) embedded submanifold of codimension 0 d 2n, then M has the structure of a C s - smooth manifold of dimension 2n d equied with the subsace toology. In addition we remark that the embedded submanifolds of codimension d = 0 are exactly the oen sets of C n. Definition 27. Let C n, we define the tangent sace of C n at as the real 2n-dimensional vector sace { } T C n = san R x 1, y 1,..., x n,. (28) y n Definition 28. Let M C n be an embedded submanifold of codimension 0 d 2n and let M. We define the tangent sace of M at to be the real (2n d)-dimensional vector sace T M = {X T C n X r j = 0 for all 1 j d}. (29) Where r : U R d is any locally defined defining function for M at. It can be shown that the definition of the tangent sace T M does not deend on the choice of locally defined defining function at. Note that we do not define the tangent sace T M to be the sace of oint-derivations at. This is done in order to avoid infinite-dimensional tangent saces. Indeed, if M is a C s -smooth manifold then the sace of oint-derivations at is finite-dimensional if and only if s =. 17

20 Definition 29. Let M C n be an embedded submanifold. We define the tangent bundle of M as the set T M = {} T M. (30) M For our alications it is sufficient to treat the tangent bundle T M as a set without any additional structure. With some work we could however have given T M the structure of a manifold. This would give us a articularly elegant formulation of vector fields on M. Instead we shall settle for the following equivalent formulation. Definition 30. Let M C n be an embedded submanifold. A C s -smooth (0 s ) vector field on M is a ma X : M T M of the form X() = (, X ). Where X = n a j () x j + b j () y j T M, (31) j=1 has coefficients a j () and b j () deending C s -smoothly on. We shall frequently identify a vector field X : M T M with the associated rojection X. Definition 31. Let V be a real vector sace. We define the comlexification of V as the comlex vector sace CV = C V. (32) In other words the comlexification CV is exactly the comlex vector sace we obtain by extending V to be a vector sace over the comlex numbers. It is clear that if V is finite-dimensional then dim R V = dim C CV. In order to better understand the comlexified tangent saces of C n we recall the Wirtinger artial differential oerators defined for 1 j n by = 1 ( z j 2 z j = 1 2 i ) x j y j ( + i x j y j (33) ). (34) The Wirtinger artial differential oerators give an alternate way of describing the comlexified tangent saces of C n. 18

21 Theorem 3.1. Let C n then { } CT C n = san C x 1, y 1,..., x n,, (35) y n or equivalently { } CT C n = san C z 1, z 1,..., z n,. (36) z n The concet of a tangent bundle of an embedded submanifold easily generalizes to comlexified tangent saces. Definition 32. Let M C n be an embedded submanifold, we define the comlexified tangent bundle of M as the set CT M = {} CT M. (37) M Definition 33. Let M C n be an embedded submanifold. A C s -smooth (0 s ) comlex vector field on M is a ma X : M T M of the form X() = (, X ). Where X CT M, and where X = n a j () x j + b j () y j, (38) j=1 has coefficients a j () and b j () deending C s -smoothly on. As with real vector fields we shall tyically identify a comlex vector field X : M T M with the associated rojection X. It is easy to verify that a ma X : M T M is a C s -smooth comlex vector field if and only if it is of the form X() = (, X ), where X = n c j () z j + d j z j CT M, (39) j=1 has coefficients c j () and d j () deending C s -smoothly on. Using the Wirtinger artial derivatives we introduce the concet of a conjugate tangent vector. Definition 34. Let C n and let X CT C n be given by X = n j=1 a j z j + b j z j. (40) 19

22 We define the conjugate tangent vector X CT C n by X = n j=1 b j z j + a j z j. (41) It is trivial to show that conjugation is an antilinear and involutive oeration. In this sense we see that conjugation of tangent vectors behaves exactly as the usual conjugation of comlex numbers. With some more work we shall show that if M C n is an embedded submanifold and M. Then for any X CT M we also have that X CT M. In order to show this we first rove the following theorem which is imortant in its own right. Theorem 3.2. Let M C n be an embedded submanifold and let M. If r : U R d is a locally defined defining function for M at, then CT M = {X CT C n X r j = 0 for all 1 j d}. (42) Proof. Let r : U R d be a locally defined defining function for M at. First suose that X CT M, then X = a j Y + ib j Y for some real tangent vector Y T M. It follows trivially that X r j = 0 for all 1 j d. Conversely suose that X CT C n is such that X r j = 0 for all 1 j d. Writing n X = (a k + ib k ) x k + (c k + id k ) y k (43) k=1 for real numbers a k, b k, c k and d k we get that X = Y +iz. Where Y, Z T C n are real tangent vectors of C n at given by Y = n k=1 a k x k + c k y k, (44) and Z = n k=1 b k x k + d k y k. (45) By assumtion X r j = Y r j + iz r j = 0 for all 1 j d, this imlies that Y, Z T M. It follows that X = Y + iz CT M. Using this theorem we can show that comlexified tangent saces are closed under conjugation. Theorem 3.3. Let M C n be an embedded submanifold and let M. If X CT M, then X CT M. 20

23 The real numbers R may be regarded as the oints z C that are unchanged by conjugation. The following theorem shows that a similar result holds for tangent saces; the real tangent sace T M is exactly the collection of comlexified tangent vectors X CT M which satisfy X = X. Theorem 3.4. Let M C n be an embedded submanifold and let M. Then T M = { X CT M X = X }. (46) In other words T M is exactly the real subsace of comlexified tangent vectors X CT M of the form X = n j=1 c j z j + c j z j. (47) Proof. First suose that X = n j=1 a j x j + b j y j T M. Define c j = a j + ib j, then a simle comutation shows that X = n j=1 c j z j + c j z j. (48) In other words X = X. Conversely, if X CT M satisfies X = X, then X is of the form n X = c j z j + c j z j. (49) j=1 Writing c j = a j + ib j where a j and b j are real, we see that X = n j=1 a j x j + b j y j. (50) Which is clearly a real tangent vector, that is X T M. 3.2 The comlex tangent sace Let M C n be an embedded submanifold of codimension 0 d 2n and let M. We may regard the tangent sace T M as a subsace of C n by identifying each tangent vector X = n j=1 a j x j + b j y j T M (51) with its corresonding geometric tangent vector (a 1 +ib 1,... a n +ib n ) C n. Under this identification we may regard T M as a real (2n d)-dimensional 21

24 subsace of C n. In general this sace does not have the structure of a comlex vector subsace of C n, but we may always decomose where HT M is the maximal comlex subsace T M = HT M X M, (52) HT M = {z T M iz T M}, (53) and where X M is the orthogonal comlement of HT M in T M. In order to make sense of this decomosition when T M is regarded as the usual algebraic tangent sace we introduce the standard smooth structure on T C n. Definition 35. Let C n, we defined the standard smooth structure on T C n as the linear ma J : T C n T C n n J a j x j + n = b j y j x j + a j y j. (54) j=1 Note that the J-oerator corresonds to multilication by the imaginary unit i in the geometric tangent sace T C n C n. This suggests the following result. Theorem 3.5. Let M C n be an embedded submanifold and let M. The tangent sace T M decomoses as T M = HT M X M where HT M T M is the comlex tangent sace j=1 HT M = {X T M J(X ) T M}, (55) and where X T M is the real art of T M given as the orthogonal comlement of HT M in T M. Similar to the definition of the tangent bundle we introduce the following notation. Definition 36. Let M C n be an embedded submanifold, we define the subsets HT M T M and XM T M by HT M = {} HT M, (56) and M XM = M {} X M. (57) We shall later see that the subsets HT M, XM T M need not be subbundles. On the other hand the decomosition T M = HT M X M clearly imlies that HT M T M is a subbundle if and only if XM T M is a subbundle. 22

25 Theorem 3.6. Let M C n be an embedded submanifold and let M. Then HT M has a basis of the form where X j T M for each 1 j m. {X 1, J(X 1 ),..., X m, J(X m )}, (58) Proof. Let X HT M be any non-zero vector, then X and J(X ) form a linearly indeendent set. The existence of a basis as above then follows from a simle induction argument. The next theorem gives an imortant bound on the dimension of the comlex tangent sace HT M. Theorem 3.7. Let M C n be an embedded submanifold of codimension 0 d 2n and let M. The dimension of the comlex tangent sace HT M T M is an even number which satisfies the inequality. 2n 2d dim HT M 2n d. (59) Proof. The fact that HT M is even-dimensional follows immediately from the revious theorem. The uer bound dim HT M 2n d is an immediate consequence of the inclusion HT M T M. In order to obtain the lower bound we write HT M = T M J(T M) and consider the quotient sace T M J(T M) T M J(T M). (60) Noting that T M J(T M) T C n we see that dim T M J(T M) T M J(T M) dim T M J(T M) dim T C n = 2n. (61) It follows that dim T M + dim J(T M) dim HT M = dim T M J(T M) T M J(T M) 2n. (62) One easily verifies that dim T M = dim J(T M) = 2n d which when combined with the above inequality easily yields the necessary lower bound. Later we shall use the above lower bound in order to define the concet of a generic CR-submanifold. It is therefore of some interest to investigate whether this is the best ossible bound for a given n N and a given 0 d 2n. Suose first that d n, then M = C n d R d C n is clearly an embedded submanifold of codimension d. Furthermore, if M then { } HT M = san x 1, y 1,...,, (63) 23 x n d y n d

26 so that dim HT M = 2n 2d. If instead n < d, then M = R 2n d {0} d n C n is an embedded submanifold of codimension d. Now if M then one easily shows that HT M is trivial, so that dim HT M = 0. This shows that the given lower bound is indeed the best ossible bound. We now turn our attention towards the comlexified comlex tangent sace CHT M. We first introduce the following useful notation. Definition 37. Let M C n be an embedded submanifold, we define the subsets CHT M CT M and CXM CT M by CHT M = {} CHT M, (64) and T (0,1) M CXM = M {} CXM. (65) We now introduce the comlex vector subsaces T (1,0) M CT M and M CT M. We then show that these saces lead to a nice decomosition of CHT M. Definition 38. Let C n, we define the comlex vector subsaces T (1,0) C n CT C n and T (0,1) C n CT C n by { } T (1,0) C n = san C z 1,...,, (66) z n and { } C n = san C z 1,...,. (67) z n T (0,1) Definition 39. Let M C n be an embedded submanifold of codimension 0 d 2n and let M. We define the comlex vector subsaces T (1,0) M CT M and T (0,1) M CT M by and T (1,0) M = CT M T (1,0) C n, (68) T (0,1) M = CT M T (0,1) C n. (69) The sace T (1,0) M is often referred to as the holomorhic tangent sace at M at, while the sace T (0,1) M is often referred to as the antiholomorhic tangent sace of M at. The following theorem gives a simle way of comuting the saces T (1,0) M and T (0,1) M. The roof is a trivial consequence of Theorem

27 Theorem 3.8. Let M C n be an embedded submanifold of codimension 0 d 2n and let M. If r : U R d is a locally defined defining function for M at, then (i) T (1,0) M = { X T (1,0) C n X r j = 0 for 1 j d }, { } (ii) T (0,1) M = X T (0,1) C n X r j = 0 for 1 j d. Theorem 3.9. Let M C n be an embedded submanifold and let M. Then the following holds (i) T (1,0) M T (0,1) M = {0}, (ii) T (0,1) M = T (1,0) M. Proof. The first roerty is clear from the fact that T (1,0) C n T (0,1) C n. In order to rove the second roerty, let 0 d 2n be the codimension of M and let r : U R d be a locally defined defining function for M at. Note from Theorem 3.8 that X = n j=1 a j z j CT M lies in T (0,1) M if and only if X r j = 0 for all 1 j d. Equivalently, using that r is real-valued we see that X lies in T (0,1) M if and only if X r j = 0 for all 1 j d. By Theorem 3.8 this is equivalent to X T (1,0) M. Definition 40. Let M C n be an embedded submanifold. We define the sets T (1,0) M CT M and T (0,1) M CT M by letting T (1,0) M = {} T (1,0) M, (70) M and letting T (0,1) M = M {} T (0,1) M. (71) We shall later see that the subsets T (1,0) M, T (0,1) M CT M need not be subbundles. Note however that from Theorem 3.9 the set T (1,0) M will be a subbundle if and only if T (0,1) M is a subbundle. Definition 41. Let M C n be an embedded submanifold. A C s -smooth (0 s ) (1, 0)-vector field over M is a C s -smooth comlex vector field of the form X : M T (1,0) M. Similarly a C s -smooth (0 s ) (0, 1)-vector field over M is a C s -smooth comlex vector field of the form X : M T (0,1) M. Let V be a real vector sace. A linear ma J : V V is called a comlex structure if J 2 (x) = x for each x V. A basis result on comlex structures says that any such ma extends to a C-linear ma J : CV CV. This immediately imlies the following. 25

28 Theorem If M, then the comlex structure ma J : T C n T C n extends to be C-linear as a ma J : CT C n CT C n. The C-linearity of the ma J : T C n T C n easily gives the following alternate descrition of the comlexified comlex tangent sace CHT M. Theorem Let M C n be an embedded submanifold and let M. Then CHT M = {X CT M J(X ) CT M}. (72) If we exress our comlexified tangent vectors in terms of the Wirtinger derivatives then the above ma J : CT C n CT C n takes the following simle form. Theorem Let C n and let X CT C n be given by X = n j=1 a j z j + b j z j, (73) then J(X ) = n ia j z j ib j z j. (74) j=1 Proof. By C-linearity ( J(X ) = 1 n ) ( ) a j J 2 x j=1 j ia j J y j (75) ( + 1 n ) ( ) b j J 2 x j=1 j + ib j J y j (76) = 1 n a j 2 y j=1 j + ia j x j (77) + 1 n b j 2 y j=1 j ib j x j (78) ( = 1 n ia j 2 x j=1 j i ) z j (79) ( + 1 n ibj 2 x j=1 j + i ) y j (80) n = ia j z j ib j z j. (81) j=1 26

29 The last statement may be rehrased in terms of the eigenvectors of the ma J : CT C n CT C n. Note that if X CT C n is an eigenvector for J corresonding to the eigenvalue λ C n, then X = J 2 (X ) = λ 2 X. This imlies that the only ossible eigenvalues for J are i and i. Now by the revious theorem we see that T (1,0) to the eigenvalue i and that T (0,1) eigenvalue i. C n is the eigensace corresonding C n is the eigensace corresonding to the Theorem Let M C n be an embedded submanifold and let M. If {X 1, J(X 1 ),..., X m, J(X m )} (82) is a basis for HT M, then {X 1 + ij(x 1 ),..., X m + ij(x m ), X 1 ij(x 1 )..., X m ij(x m )} (83) is a basis for CHT M. Consequentially CHT M slits as CHT M = T (1,0) M T (0,1) M. (84) Proof. Let {X 1, J(X 1 ),..., X m, J(X m )} be a basis for HT M (from Theorem 3.6 a basis of this form always exists). By Theorem 3.4 we may find comlex numbers c j,k for each 1 j n and 1 k m so that X k = c j,k z j=1 j + c j,k z j. (85) Using the revious theorem we see that X k ij(x k ) = X k + ij(x k ) = n j=1 n j=1 c j,k c j,k z j, (86) z j. (87) Note that X k ij(x k ) T (1,0) M and that X k +ij(x k ) T (0,1) M. Moreover a simle comutation shows that {X 1 + ij(x l ),..., X m + ij(x m ), X 1 ij(x 1 ),..., X m ij(x m )} (88) is a basis for CHT M which finishes the roof. In fact the above roof clearly shows that if X HT M, then we have that X ij(x ) T (1,0) M, and that X + ij(x ) T (0,1) M. This imlies the following alternate characterization of the saces T (1,0) M and T (0,1) M. 27

30 Theorem Let M C n be an embedded submanifold and let M. Then (i) T (1,0) M = {X ij(x ) X HT M}, (ii) T (0,1) M = {X + ij(x ) X HT M}. Proof. Recall from Theorem 3.4 that if X T M, then X = X. It follows that {X ij(x ) X HT M} = {X + ij(x ) X HT M}. (89) Now from Theorem 3.9 we know that T (1,0) M = T (0,1) M, it is therefore sufficient to show that T (0,1) M = {X + ij(x ) X HT M}. (90) By the revious theorem there exists a basis for T (0,1) M of the form {X 1 + ij(x 1 ),..., X m + ij(x m )}, (91) where X j HT M for 1 j m. We define the set D (0,1) M = {X + ij(x ) X HT M}, (92) and note the obvious inclusion T (0,1) M D (0,1) M. By the remarks following the last theorem we also obtain the inclusion D (0,1) M T (0,1) M. This finishes the roof. We finish off this technical section with one final result. This result gives an alternate descrition of the comlex tangent sace HT M in terms of the sace T (1,0) M. Theorem Let M C n be an embedded submanifold and let M. The comlex tangent sace HT M T M is given by { } HT M = X + X CT M X T (1,0) M. (93) Proof. We first show that if X T (1,0) M, then X + X HT M. Now by the revious theorem we may write X = Y ij(y ) for some Y HT M. It follows from an alication of T heorem 3.9 that X + X = 2Y. This imlies that X + X HT M. Conversely if Y HT M, then we know from the revious theorem that X = 1 2 (Y ij(y )) T (1,0) M. Now Y = X + X which finishes the roof. 28

31 3.3 CR submanifolds We have already hinted at the existence of submanifolds M C n for which the subset HT M T M is not a subbundle. We are now going to show one such manifold. Let r : C 2 R 2 be the function r(x 1 + iy 1, x 2 + iy 2 ) = (x 1, y 1 + x 2 y 2 2 1), (94) and define M = { z C 2 r(z) = 0 }. First note that r is clearly C - smooth, furthermore a simle calculation shows that df 1 df 2 = dx 1 dy 1 + y 2 2 dx 1 dx 2 + 2x 2 y 2 dx 1 dy 2. (95) Clearly this does not vanish anywhere and so we see that M is a C -smooth submanifold of codimension d = 2. We now calculate the tangent sace T M at a oint = (x 1 + iy 1, x 2 + iy 2 ) M. The tangent sace is given by the collection of tangent vectors X = 2 j=1 a j x j + b j y j T C 2 which satisfy the following system of equations X f 1 = a 1 = 0, (96) X f 2 = b 1 + 2b 2 x 2 y 2 + a 2 y 2 2 = 0. (97) Solving this system we see that { } T M = san x 2 y2 2 y 2, y 2 2x 2 y 2. (98) y 1 Now if = (i, 0) then the tangent sace takes the simle form { } T M = san x 2,. (99) y 2 Recalling our original definition of the comlex tangent sace HT M we immediately see that dim HT M = 2. Now if instead = (0, 1/ i/ 3 3), then T M is given by { ( ) T M = san 1 2 ( ) x y 2, 1 2 } y 2 2. (100) 3 3 y 1 Now suose that X T M satisfies J(X ) T M. Writing X = a ( ) 1 2 x y 2 + b ( ) 1 2 y y 1, (101) 29

32 we see that ( ( ) 1 2 J(X ) = a 3 3 x 2 + ) ( ( ) 1 2 y 2 + b x 1 ) y 2. (102) By assumtion J(X ) T M, now T M contains no artial derivatives with resect to the x 1 -variable and so we may immediately conclude that b = 0. It follows that J(X ) is of the form ( ( ) 1 2 J(X ) = a 3 3 x 2 + ), (103) y 2 which is clearly not in T M unless a = 0. We must therefore conclude that HT M is trivial, that is dim HT M = 0. We give embedded submanifolds M C n for which HT M T M is a subbundle a secial name. Definition 42. Let M C n be an embedded submanifold. We say that M is a CR-submanifold if we for each air of oints, q M have that dim HT M = dim HT q M. In this case we refer to the common dimension of the comlex tangent saces as the CR-dimension of M which we denote by CR dim M. For the remaining chaters we shall work exclusively with CR-submanifolds. We shall see that the CR-submanifolds include many of the most imortant submanifolds in analysis. The following result gives several equivalent definitions for a submanifold to be a CR-submanifold. The theorem follows easily from all our revious work. Theorem Let M C n be an embedded submanifold. The following are equivalent. (i) M is a CR-submanifold. (ii) HT M T M is a subbundle. (iii) XM T M is a subbundle. (iv) CHT M CT M is a subbundle. (v) CXM CT M is a subbundle. (vi) T (1,0) M CT M is a subbundle. (vii) T (0,1) M CT M is a subbundle. 30