In this paper, we classify all nonconstant smooth CR maps from a sphere with n > 3 to the Shilov boundary of a bounded symmetric domain of Cartan type I under the condition that p − q < 3 n − 4. We show that they are either linear maps up to automorphisms of S _n,1 and S_p,q or D’Angelo maps. This is the first classification of CR maps into the Shilov boundary of bounded symmetric domains other than sphere that includes nonlinear maps. The rigidity of holomorphic maps between open pieces of a sphere was first studied by Poincaré [13] in 2-dimensional case and later by Alexander [1] and Chern and Moser [2] for general dimensions. Then Webster [16] obtained rigidity for holomorphic maps between open pieces of spheres of different dimension, proving that any such map between spheres in and extends as a totally geodesic map between balls with respect to the Bergman metric. Later, Huang [6] generalized Webster’s result for CR maps between open pieces of spheres in and under the assumption n′ − 1 < 2( n − 1). Beyond this bound, the rigidity fails as illustrated by the Whitney map. Unit ball is a bounded symmetric domain of Cartan type I with rank 1 and sphere is its Shilov boundary. However, comparing with rigidity of holmorphic maps between spheres mentioned above, holomorphic rigidity for maps between bounded symmetric domains D and D′ of higher rank remains much less understood. If the rank r′ of D′ does not exceed the rank r of D and both ranks r , r′ ≥ 2, the rigidity of proper holomorphic maps f : D → D′ was conjectured by Mok [12] and proved by Tsai [15] , showing that f is necessarily totally geodesic (with respect to the Bergmann metric). For the case r < r′ , in [11] , Zaitsev and author showed the rigidity of CR maps f : S_p,q → S_p′,q′ under the assumption that q ≥ 2 and ( p′ − q′ ) < 2( p − q ). Here, S_p,q and S_p′,q′ are the Shilov boundaries of a bounded symmetric domains of Cartan type I((See §1 for definition) and q and q′ are the ranks of S_p,q and S_p′,q′ , respectively. When ( p′ − q′ ) = 2 ( p − q ), then the rigidity fails to hold, as authors introduced the generalized Whitney map as a counterexample in the same paper. Recently, in [14] , A. Seo introduced a nonlinearizable proper holomorphic maps between S_p,q and S _{2p−1,2q−1} . Therefore, to classify all CR maps between S_p,q and S_p′,q′ when p′ − q′ ≥ 2( p − q ), one should consider nonlinear maps. In [9] , Huang, Ji and Xu classified all locally defined CR maps between S _n,1 and S _n′,1 under the assumption that 3 < n ≤ n′ < 3 n − 3. It is proved that such map is either a linear map or a D’Angelo map. In this paper, we generalize the result of Huang, Ji and Xu. We define D’Angelo map from a sphere into the Shilov boundary of bounded symmetric domains of type I as follows: Definition 1.1. Let be the set of all complex p × q matrices. A map f_θ : S _n,1 → S_p,q for a fixed 0 < θ ≤ π /2, is called a D’Angelo map if f_θ is equivalent to the following map up to automorphisms of S _n,1 and S_p,q , where W_θ ( z ) is a map from S _n,1 to S _3n−3,1 defined by and I _q−1 is the identity matrix of size ( q −1). This map is not linear after composing with any automorphisms of S _n,1 and S_p,q . For q = 1 and θ = π /2, this is the classical Whitney map between unit balls in and respectively. In this paper, we classify all locally defined CR maps from a sphere S _n,1 with n > 3 into the Shilov boundary S_p,q of a general Cartan type I bounded symmetric domain of higher rank. We showed Theorem 1.2. Let f be a nonconstant smooth CR map from an open piece of S _n,1 into S_p,q . Assume that n > 3 and p − q < 3 n − 4. Then after composing with suitable automorphisms of S _n,1 and S_p,q , f is either a linear embedding or D’Angelo map . Note that our basic assumption p − q < 3 n −4 corresponds precisely to the optimal bound n′ − 1 < 3( n − 1) in the rank 1 case ( q = 1) of maps between spheres, where n −1 and n′ − 1 are the CR dimensions of the spheres. Throughout this paper we adopt the Einstein summation convention unless mentioned otherwise. In this section, we review CR structure and Grassmannian frames adapted to S_p,q . For details, we refer [2] and [11] as references. In this section, we let Greek indices α , β , γ , … and Latin indices j , k , ℓ , … run over {1,…, q } and {1,…, p − q }, respectively. For q = 1, i.e., sphere case, we omit Greek indices. A Hermitian symmetric domain D_p,q of Cartan type I has a standard realization in the space of p × q matrices, given by where I_q is the q × q identity matrix and The Shilov boundary of D_p,q is given by In particular, S_p,q is a CR manifold of CR dimension ( p − q ) × q . For q = 1, S _p,1 is the unit sphere in . We shall always assume p > q so that S_p,q has positive CR dimension. Let Aut ( S_p,q ) be the Lie group of all CR automorphisms of S_p,q . By [10, Theorem 8.5], every f ∈ Aut ( S_p,q ) extends to a biholomorphic automorphism of the bounded symmetric domain D_p,q . Consider the standard linear inclusion Then we may regard S_p,q as a real submanifold in the Grassmanian Gr ( q, p + q ) of all q -planes in and Aut ( S_p,q )(= Aut ( D_p,q )) becomes a subgroup of the automorphism group of Gr ( q, p + q ). For column vectors u = ( u ₁ , … , u _p+q ) ^t and v = ( v ₁ , … , v _p+q ) ^t in , define a Hermitian inner product by A Grassmannian frame adapted to S_p,q , or simply S_p,q-frame is a frame { Z ₁ , …, Z _p+q } of with det ( Z ₁ , …, Z _p+q ) = 1 such that scalar product <·, ·> in basis ( Z ₁ , …, Z _p+q ) is given by the matrix Now let be the set of all S_p,q -frames. Then is identified with SU ( p, q ) by the left action. The Maurer-Cartan form on is given by the equation where π satisfies the trace-free condition and the structure equation where the capital Greek indices Λ, Γ, Ω etc. run from 1 to p + q . From now, we will use the notation Z := ( Z ₁ , … , Z _q ), X = ( X ₁ , … , X _p−q ) := ( Z _q+1 , … , Z _p ), Y = ( Y ₁ , … , Y _q ) := ( Z _p+1 …, Z_p+q ) so that the Maurer-Cartan form with respect to the basis ( Z, X, Y ) can be written as with the symmetry relations By abuse of notation, we also denote by Z the q -dimensional subspace of spanned by Z ₁ , …, Z_q . Then the defining equations of S_p,q can be written as and hence their differentiation yields By substituting into (1, 0) component of (2.3) we obtain, in particular, when restricted to the (1, 0) tangent space. Comparing the dimensions, we conclude that span the space of contact forms on S_p,q , i.e., is the complex tangent space of S_p,q . The structure equation is given by Moreover, since we conclude that θ_α^j form a basis in the space of (1, 0) forms. There are several types of frame changes. Definition 2.1. We call a change of frame Finally, we shall use the change of frame given by such that and where The new frame is an S_p,q -frame and the related 1-forms remain the same, while change to Let f : S _n,1 → S_p,q be a (germ of a) smooth CR mapping. We shall identify S _n,1 and its image f ( S _n,1 ) ⊂ S_p,q . We consider the connection forms φ , θ^j , 𝜓 , ω_j^k , σ_j , ξ with j, k = 1, …, n − 1 on S _n,1 and denote by capital letters Φ _α^β , Θ _α^j , Ψ _α^β , Ω _J^K ,Σ _K^β , Ξ _α^β with α, β = 1,…, q and J, K = 1,…, p − q , their corresponding counterparts on S_p,q . We also define one forms φ_α^β , θ_α^J adapted to f as follows: Definition 3.1. We say that f is of contact rank r if f sends any nonzero vector in TS _n,1 / T^cS _n,1 to a rank r vector in TS_p,q / T^cS_p,q . For a map f of contact rank r , we define φ_α^β , θ_α^J for α = 1, …, q and J = 1, …, p − q adapted to f by and 0 otherwise. In this section we show the following lemma. Lemma 3.2. For any nonconstant local CR map f : S _n,1 → S_p,q with p − q < 3( n −1), there exist r ∈ {1, 2} and a choice of S_p,q-frames such that f is of contact rank r and the forms φ_α^β , θ_α^J adapted to f satisfy Proof is a slight modification of the proof of Lemma 4.2 and argument in §.5 of [11] . We refer [11] for details. Proof . Since φ and Φ = (Φ _α^β ) are contact forms on S _n,1 and S_p,q , respectively, the pull back of Φ f is a span of φ . Choose a diagonal contact form of S_p,q and say Φ ₁ ¹ . Then we can write for some smooth function λ. At generic points, we may assume that either λ ≡ 0 or λ never vanishes. By differentiating (3.2) and using (2.4) we obtain Arguing similar to [11] we conclude λ ≥ 0 and, after dilation of Φ ₁ ¹ , we may assume that λ = 1 if λ ≢ 0. Suppose that Φ _α^α vanishes identically for all α . Then we obtain Since each Θ _α^J is a (1, 0) form, it follows that i.e., f ( S _n,1 ) is a totally real submanifold. Since S _n,1 is Levi-nondegenerate, this implies that f is a constant map, which contradicts our assumption. Hence there exists at least one diagonal term of Φ whose pullback does not vanish identically. Choose such a diagonal term of Φ, say Φ ₁ ¹ . Then (3.3) yields Therefore after a suitable rotation of S_p,q , we may assume that Write for some smooth functions λ _α . Then by differentiating (3.6) and using (2.4) together with (3.4), (3.5), we obtain Choose a suitable change of position that leaves Θ ₁ ^J invariant and replaces Θ _α^J with Θ _α^J − λ _α Θ ₁ ^J for α ≥ 2. This change of position leaves Φ ₁ ¹ invariant and transforms Φ _α ¹ into Φ _α ¹ − λ _α Φ ₁ ¹ for α ≥ 2. After performing such change of position, (3.6) becomes and (3.7) becomes Sincec Θ _α^j are (1, 0) but θ_j are (0, 1) and linearly independent, it follows that Next for each α ≥ 2, let for another smooth function λ _α . If λ _α ≡ 0 for all α ≥ 2, then by differentiation, we obtain which yields In this case, by considering the differentiation of and substituting (3.10), we conclude that which implies that df ( T ) modulo T^cS_p,q is a rank 1 vector for any T ∈ TS _n,1 transversal to T^cS _n,1 . That is to say, f is of contact rank 1 and the forms adapted to f satisfy Suppose there exists α such that λ _α ≢ 0. We may assume α = 2. After a dilation of Φ ₂ ² , we may assume that at generic points, λ ₂ = 1. By differentiating (3.9) for α = 2 and substituting (3.8) we obtain Hence after a suitable rotation where ( U_K^J ) is unitary matrix leaving Θ _α^j , j = 1,…, n −1, invariant, we may assume that and otherwise. Write for some smooth function λ _α . Then as before, we can choose a suitable change of position that leaves Θ ₁ ^J and Θ ₂ ^J invariant and replaces Θ _α ^J with Θ _α ^J − λ _α Θ ₂ ^J for α > 2, which also leaves Φ ₁ ¹ , Φ ₂ ¹ and Φ ₂ ² invariant and transforms Φ _α ² into Φ _α ² − λ _α Φ ₂ ² for α > 2. By (3.8), after performing such change of position, the following property still holds and (3.11) becomes By differentiating this we obtain which yields Write for some smooth functions λ _α . Suppose that λ _α ≡ 0 for all α . Then as before, we can obtain i.e., f is of contact rank 2 and the forms adapted to f satisfy Suppose there exists α such that λ _α ≠ 0 We may assume α = 3. After a dilation of Φ ₃ ³ , we may assume that at generic points, λ ₃ = 1, i.e., By differentiating this, we obtain hen by (3.8) and (3.12), we have at most p − q − 2( n −1) linearly independent (1, 0) forms on the left-hand side, while on the right-hand side we have n − 1 linearly independent (1, 0) forms. Since we assumed that p − q < 3( n − 1), this is a contradiction. Next we will show that there exists a choice of frames such that Write for some η_α^J . Consider the equations obtained by differentiating (3.13): where and Let α > r . Then left-hand side of (3.14) contains at most one (1, 0) form, while the right-hand side contains ( n − 1) linearly independent (1, 0) forms with n − 1 > 1 unless η_α^J = 0. Therefore we conclude that or equivalently Finally, define a matrix ( B_α^J ) by where η_α^J satisfies Consider the change of frame of S_p,q discussed after Definition 2.1, given by such that and A_α^β satisfies Since the sum here is hermitian, one can always choose A_α^β with this property. Then Φ _α^β remain the same while change to Therefore the new Θ _α ^J satisfies In this section, we determine second fundamental forms given by Ω _J^K . Then we determine Ψ _α^β and Σ _α^J . By using these forms, we construct a linear subspace of Gr ( q, p+q ) that contains the image of a given embedding(Lemma 4.1, Lemma 4.2). Their proofs are slight modification of the proof of Proposition 7.1 in [11] . Let f be a CR map of contact rank r with r ∈ {1, 2}. Differentiate (3.1) using the structure equations to obtain where and 0 otherwise. 4.1. Contact rank 1 map Choose α > 1 and J = j . Then (4.1) takes the form By Cartan Lemma we obtain for fixed j . Since Ψ is independent of j = 1, … , n −1 and we assumed n −1 > 1, we obtain We will show the following lemma. Lemma 4.1. There exists ( p−q +2)- dimensional subspace V ₁ and ( q −1)- dimensional subspace V ₂ in orthogonal to each other such that Gr (1, V ₁ ) ⊕ V ₂ contains the image f ( S _n,1 ). Proof . Choose an open set M ⊂ S _n,1 where f is defined. Let Z , X , Y be constant vector fields of forming a S_p,q -frame at a fixed reference point of f ( M ) and let be an adapted S_{p, q} -frame along f ( M ). Write so that (4.3) - (4.5) take the form Since Z, X, Y form an adapted frame at a reference point of M , we may assume that at the reference point. Since Z, X, Y are constant vector fields, i.e., dZ = dX = dY = 0, differentiating (4.6) and using (2.1) we obtain Next, it follows from Lemma 3.2 and (4.2) that in particular, the span of is independent of the point in M . Hence together with (4.3) and (4.7), we conclude Furthermore, (4.8) for α = 1 together with Lemma 3.2 and (4.2) (and with the symmetry relations analogous to (2.2)) we obtain Now with (4.9) taken into account, (4.10) becomes Thus each of the vector valued functions for a fixed β satisfies a complete system of linear first order differential equations. Then by the initial condition (4.7) and the uniqueness of solutions, we conclude, in particular, that Hence (4.3) implies Now setting we still have whereas (4.11) becomes implying Then we conclude that where 4.2. Contact rank 2 map Choose α > 2 and J = j or J = n − 1 + j . Then (4.1) takes the form Since Ψ is independent of j = 1,…, n − 1 and we assumed n − 1 > 1, by Cartan Lemma we obtain Use (4.1) for either α = 1 and J = n − 1 + J or α = 2 and J = j or α = 1, 2 and J > 2( n −1) to obtain By Cartan’s Lemma, we obtain where θ is an ideal generated by θ ¹ ,…, θ ⁿ⁻¹ . Since by using (4.16), we conclude that Moreover, since Ψ is independent of j , substituting (4.18) into (4.13) and (4.14), we obtain Next we will determine second fundamental forms of f as in [16] . We will show that it has a trivial solution only. For details, we refer [16] . Use (4.1) for α = 1 and J = j ≤ ( n − 1) to obtain Then by Cartan Lemma, we obtain By symmetry relation for ­Ω, we obtain Furthermore, differentiation of by using the structure equations yields or equivalently Therefore we obtain for some pure imaginary function g . Similar computation for (4.1) with α = 2 and J = n − 1 + j together with the relation yields and for a pure imaginary function h . Take a real vector change of S_p,q defined by for a smooth function μ satisfying in (4.19) and fixing the rest. Then after the frame change, we obtain By differentiating (4.20),(4.21),(4.22) and substituting (4.12) and (4.19), we obtain Then by Cartan Lemma, we obtain By (4.16) and symmetry relation for Σ, we obtain Now let Then (4.1) implies Write Differentiate (4.24) and substitute (4.18) to obtain which implies If p−q < 3( n −1), then (4.26) has trivial solution only.(See [3] .) Therefore we obtain or equivalently Similar computation for using (4.25) yields By (4.18) and (4.27), we can write By differentiating this, we obtain By (4.17) and (4.23) we can show that the left-hand side of (4.28) contains at most one (0, 1) form, while the right-hand side contains ( n − 1) linearly independent (0, 1) forms unless η_k^J = 0. Hence we conclude that or equivalently and therefore by substituting (4.17) and (4.23) into (4.28), we obtain Similar computation for implies Furthermore, by substituting (4.29) to(4.15) with J = j , we obtain Finally we will determine Ψ and Σ. By (4.19), we can write By differentiating this and substituting (4.12) and (4.19), we obtain By (4.30), this implies or equivalently Let By differentiation, we obtain which yield or equivalently Since Ξ ₁ ² is independent of j , we obtain Similar computation for Σ ₂ ^J yields Summing up we obtain the following: For any contact rank 2 local CR embedding f from S _n,1 into S_p,q , there is a choice of frames such that We will show the following lemma. Lemma 4.2. There exist ( n +1)- dimensional subspaces V ₁ , V ₂ and ( q −2)- dimensional subspace V ₃ in orthogonal to each other such that Gr (1, V ₁ ) ⊕ Gr (1, V ₂ ) ⊕ V ₃ contains the image f ( S _n,1 ). Proof . We use the same method in Lemma 4.1. Let M ⊂ S _n,1 , Z, X, Y and be as in Lemma 4.1. It follows from Lemma 3.2 and (4.31) that in particular, the span of is independent of the point in M . Hence as in Lemma 4.1, we conclude Furthermore, (4.8) implies In particular, restricting to α = 1 and J = j ≤ n with (4.31)-(4.34) and (4.37) taken into account, we obtain Repeating the above argument for λ and ζ instead of η , we obtain and Thus each of the vector valued functions for a fixed K and for a fixed β satisfies a complete system of linear first order differential equations. Then as in Lemma 4.1 we conclude, in particular, that and Hence (4.35) implies Similar computation for implies Now setting we still have whereas (4.38), (4.39) become implying Then together with (4.36) we conclude that where Suppose f is of contact rank 1. Then by Lemma 4.1, there exist ( p − q + 2)-dimensional subspace V ₁ and ( q − 1)-dimensional subspace V ₂ such that the image of f is contained in Gr (1, V ₁ ) ⊕ V ₂ . The V ₂ -component of f is a constant map. Therefore it is enough to show that Gr (1, V ₁ )-component of f is either a linear map or Whitney map. But Gr (1, V ₁ ) = . Therefore by the result of [9] under the condition n > 3 and ( p − q ) < 3 n − 4, we conclude that Gr (1, V ₁ )-component of f is either a flat embedding or D’Angelo map. Suppose f is of contact rank 2, then by Lemma 4.2, there exist ( n +1)-dimensional subspaces V ₁ , V ₂ and ( q − 2)-dimensional subspace V ₃ such that the image of f is contained in Gr (1, V ₁ )⊕ Gr (1, V ₂ )⊕ V ₃ . As before, it is enough to show that Gr (1, V ₁ ) and Gr (1, V ₂ )-components of f are linear. Since V ₁ and V ₂ are of dimension ( n +1), each component of f is a CR automorphism of S _n,1 . Therefore, it is projective linear, which completes the proof.