Let $M^{n}(c)$ be an $n$-dimensional space form with constant sectional curvature $c$. Alencar-do Carmo-Tribuzy [5] classified all parallel mean curvature (abbrev. PMC) surfaces with non-negative Gaussian curvature $K$ in $M^n(c) \times \mathbb{R}$ with $c \lt 0$. Later on, Fetcu-Rosenberg [28] generalized their results for $c \neq 0$. However, the classification to PMC surfaces in $M^n(c) \times \mathbb{R}$ with $K \equiv 0$ is still open. In this paper, we give a complete classification to the PMC surfaces in $M^n(c) \times \mathbb{R}$ with $K \equiv 0$ whose tangent plane spans the constant angle with factor $\mathbb{R}$.
References
U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surfaces in $\mb{S}^{2} \times \mb{R}$ and $\mb{H}^{2} \times \mb{R}$, Acta Math. 193 (2004), no. 2, 141–174. 10.1007/bf02392562 MR2134864 1078.53053 euclid.acta/1485891698
U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surfaces in $\mb{S}^{2} \times \mb{R}$ and $\mb{H}^{2} \times \mb{R}$, Acta Math. 193 (2004), no. 2, 141–174. 10.1007/bf02392562 MR2134864 1078.53053 euclid.acta/1485891698
H. Alencar, M. do Carmo, I. Fernández and R. Tribuzy, A theorem of H. Hopf and the Cauchy-Riemann inequality II, Bull. Braz. Math. Soc. (N.S.) 38 (2007), no. 4, 525–532. 10.1007/s00574-007-0058-x MR2371943 1157.53349 H. Alencar, M. do Carmo, I. Fernández and R. Tribuzy, A theorem of H. Hopf and the Cauchy-Riemann inequality II, Bull. Braz. Math. Soc. (N.S.) 38 (2007), no. 4, 525–532. 10.1007/s00574-007-0058-x MR2371943 1157.53349
H. Alencar, M. do Carmo and R. Tribuzy, A theorem of Hopf and the Cauchy-Riemann inequality, Comm. Anal. Geom. 15 (2007), no. 2, 283–298. 10.4310/cag.2007.v15.n2.a3 MR2344324 1134.53031 H. Alencar, M. do Carmo and R. Tribuzy, A theorem of Hopf and the Cauchy-Riemann inequality, Comm. Anal. Geom. 15 (2007), no. 2, 283–298. 10.4310/cag.2007.v15.n2.a3 MR2344324 1134.53031
––––, A Hopf theorem for ambient spaces of dimensions higher than three, J. Differential Geom. 84 (2010), no. 1, 1–17. MR2629507 1197.53073 10.4310/jdg/1271271791 euclid.jdg/1271271791
––––, A Hopf theorem for ambient spaces of dimensions higher than three, J. Differential Geom. 84 (2010), no. 1, 1–17. MR2629507 1197.53073 10.4310/jdg/1271271791 euclid.jdg/1271271791
––––, Surfaces of $M^2_k \times \mb{R}$ invariant under a one-parameter group of isometries, Ann. Mat. Pura Appl. (4) 193 (2014), no. 2, 517–527. 10.1007/s10231-012-0288-4 ––––, Surfaces of $M^2_k \times \mb{R}$ invariant under a one-parameter group of isometries, Ann. Mat. Pura Appl. (4) 193 (2014), no. 2, 517–527. 10.1007/s10231-012-0288-4
C. P. Aquino, H. F. de Lima and E. A. Lima, Jr., On the angle of complete CMC hypersurfaces in Riemannian product spaces, Differential Geom. Appl. 33 (2014), 139–148. 10.1016/j.difgeo.2014.02.006 MR3183370 1325.53073 C. P. Aquino, H. F. de Lima and E. A. Lima, Jr., On the angle of complete CMC hypersurfaces in Riemannian product spaces, Differential Geom. Appl. 33 (2014), 139–148. 10.1016/j.difgeo.2014.02.006 MR3183370 1325.53073
J. O. Baek, Q.-M. Cheng and Y. J. Suh, Complete space-like hypersurfaces in locally symmetric Lorentz spaces, J. Geom. Phys. 49 (2004), no. 2, 231–247. 10.1016/s0393-0440(03)00090-1 1078.53055 J. O. Baek, Q.-M. Cheng and Y. J. Suh, Complete space-like hypersurfaces in locally symmetric Lorentz spaces, J. Geom. Phys. 49 (2004), no. 2, 231–247. 10.1016/s0393-0440(03)00090-1 1078.53055
M. Batista, Simons type equation in $\mb{S}^{2} \times \mb{R}$ and $\mb{H}^{2} \times \mb{R}$ and applications, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 4, 1299–1322. 10.5802/aif.2641 M. Batista, Simons type equation in $\mb{S}^{2} \times \mb{R}$ and $\mb{H}^{2} \times \mb{R}$ and applications, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 4, 1299–1322. 10.5802/aif.2641
M. Batista, M. P. Cavalcante and D. Fetcu, Constant Mean Curvature Surfaces in $\mb{M}^2(c) \times \mb{R}$ and Finite Total Curvature, arXiv:1402.1231. 1402.1231 MR3708005 10.1007/s12220-017-9795-2 M. Batista, M. P. Cavalcante and D. Fetcu, Constant Mean Curvature Surfaces in $\mb{M}^2(c) \times \mb{R}$ and Finite Total Curvature, arXiv:1402.1231. 1402.1231 MR3708005 10.1007/s12220-017-9795-2
M. do Carmo, Some recent developments on Hopf's holomorphic form, Results Math. 60 (2011), no. 1-4, 175–183. 10.1007/s00025-011-0150-9 1256.53006 M. do Carmo, Some recent developments on Hopf's holomorphic form, Results Math. 60 (2011), no. 1-4, 175–183. 10.1007/s00025-011-0150-9 1256.53006
M. do Carmo and I. Fernández, A Hopf theorem for open surfaces in product spaces, Forum Math. 21 (2009), no. 6, 951–963. 10.1515/forum.2009.047 MR2574142 1188.53072 M. do Carmo and I. Fernández, A Hopf theorem for open surfaces in product spaces, Forum Math. 21 (2009), no. 6, 951–963. 10.1515/forum.2009.047 MR2574142 1188.53072
B.-Y. Chen, Riemannian submanifolds: A Survey, in: Handbook of Differential Geometry, Vol. I, 187–418, North-Holland, Amsterdam, 2000. 10.1016/s1874-5741(00)80006-0 B.-Y. Chen, Riemannian submanifolds: A Survey, in: Handbook of Differential Geometry, Vol. I, 187–418, North-Holland, Amsterdam, 2000. 10.1016/s1874-5741(00)80006-0
H. Chen, G. Y. Chen and H. Z. Li, Some pinching theorems for minimal submanifolds in $\mb{S}^{m}(1) \times \mb{R}$, Sci. China Math. 56 (2013), no. 8, 1679–1688. 10.1007/s11425-012-4556-y MR3079823 H. Chen, G. Y. Chen and H. Z. Li, Some pinching theorems for minimal submanifolds in $\mb{S}^{m}(1) \times \mb{R}$, Sci. China Math. 56 (2013), no. 8, 1679–1688. 10.1007/s11425-012-4556-y MR3079823
S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195–204. 10.1007/bf01425237 MR431043 0349.53041 S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195–204. 10.1007/bf01425237 MR431043 0349.53041
S. S. Chern, On surfaces of constant mean curvature in a three-dimensional space of constant curvature, in: Geometric Dynamics (Rio de Janeiro, 1981), 104–108, Lecture Notes in Math. 1007, Springer, Berlin, 1983. 10.1007/bfb0061413 S. S. Chern, On surfaces of constant mean curvature in a three-dimensional space of constant curvature, in: Geometric Dynamics (Rio de Janeiro, 1981), 104–108, Lecture Notes in Math. 1007, Springer, Berlin, 1983. 10.1007/bfb0061413
S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, in: Functional Analysis and Related Fields, 59–75, Springer, New York, 1970. 10.1007/978-3-642-49908-1_2 0216.44001 S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, in: Functional Analysis and Related Fields, 59–75, Springer, New York, 1970. 10.1007/978-3-642-49908-1_2 0216.44001
B. Daniel, Isometric immersions into $\mb{S}^{n} \times \mb{R}$ and $\mb{H}^{n} \times \mb{R}$ and applications to minimal surfaces, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6255–6282. 10.1090/s0002-9947-09-04555-3 MR2538594 1213.53075 B. Daniel, Isometric immersions into $\mb{S}^{n} \times \mb{R}$ and $\mb{H}^{n} \times \mb{R}$ and applications to minimal surfaces, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6255–6282. 10.1090/s0002-9947-09-04555-3 MR2538594 1213.53075
F. Dillen, J. Fastenakels and J. Van der Veken, Surfaces in $\mb{S}^2 \times \mb{R}$ with a canonical principal direction, Ann. Global Anal. Geom. 35 (2009), no. 4, 381–396. 10.1007/s10455-008-9140-x F. Dillen, J. Fastenakels and J. Van der Veken, Surfaces in $\mb{S}^2 \times \mb{R}$ with a canonical principal direction, Ann. Global Anal. Geom. 35 (2009), no. 4, 381–396. 10.1007/s10455-008-9140-x
F. Dillen, J. Fastenakels, J. Van der Veken and L. Vrancken, Constant angle surfaces in $\mb{S}^2 \times \mb{R}$, Monatsh. Math. 152 (2007), no. 2, 89–96. 10.1007/s00605-007-0461-9 F. Dillen, J. Fastenakels, J. Van der Veken and L. Vrancken, Constant angle surfaces in $\mb{S}^2 \times \mb{R}$, Monatsh. Math. 152 (2007), no. 2, 89–96. 10.1007/s00605-007-0461-9
F. Dillen and M. I. Munteanu, Constant angle surfaces in $\mb{H}^2 \times \mb{R}$, Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 1, 85–97. 10.1007/s00574-009-0004-1 MR2496114 1173.53012 F. Dillen and M. I. Munteanu, Constant angle surfaces in $\mb{H}^2 \times \mb{R}$, Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 1, 85–97. 10.1007/s00574-009-0004-1 MR2496114 1173.53012
F. Dillen, M. I. Munteanu and A. I. Nistor, Canonical coordinates and principal directions for surfaces in $\mb{H}^2 \times \mb{R}$, Taiwanese J. Math. 15 (2011), no. 5, 2265–2289. MR2880404 1241.53010 10.11650/twjm/1500406434 euclid.twjm/1500406434
F. Dillen, M. I. Munteanu and A. I. Nistor, Canonical coordinates and principal directions for surfaces in $\mb{H}^2 \times \mb{R}$, Taiwanese J. Math. 15 (2011), no. 5, 2265–2289. MR2880404 1241.53010 10.11650/twjm/1500406434 euclid.twjm/1500406434
J. M. Espinar, J. A. Gálvez and H. Rosenberg, Complete surfaces with positive extrinsic curvature in product spaces, Comment. Math. Helv. 84 (2009), no. 2, 351–386. 10.4171/cmh/165 MR2495798 1166.53040 J. M. Espinar, J. A. Gálvez and H. Rosenberg, Complete surfaces with positive extrinsic curvature in product spaces, Comment. Math. Helv. 84 (2009), no. 2, 351–386. 10.4171/cmh/165 MR2495798 1166.53040
J. M. Espinar and H. Rosenberg, Complete constant mean curvature surfaces and Bernstein type theorems in $M^2 \times \mb{R}$, J. Differential Geom. 82 (2009), no. 3, 611–628. MR2534989 1180.53062 10.4310/jdg/1251122547 euclid.jdg/1251122547
J. M. Espinar and H. Rosenberg, Complete constant mean curvature surfaces and Bernstein type theorems in $M^2 \times \mb{R}$, J. Differential Geom. 82 (2009), no. 3, 611–628. MR2534989 1180.53062 10.4310/jdg/1251122547 euclid.jdg/1251122547
––––, Complete constant mean curvature surfaces in homogeneous spaces, Comment. Math. Helv. 86 (2011), no. 3, 659–674. 10.4171/cmh/237 MR2803856 1227.53068 ––––, Complete constant mean curvature surfaces in homogeneous spaces, Comment. Math. Helv. 86 (2011), no. 3, 659–674. 10.4171/cmh/237 MR2803856 1227.53068
M. J. Ferreira and R. Tribuzy, Parallel mean curvature surfaces in symmetric spaces, Ark. Mat. 52 (2014), no. 1, 93–98. 10.1007/s11512-012-0170-z 1308.53092 M. J. Ferreira and R. Tribuzy, Parallel mean curvature surfaces in symmetric spaces, Ark. Mat. 52 (2014), no. 1, 93–98. 10.1007/s11512-012-0170-z 1308.53092
D. Fetcu, C. Oniciuc and H. Rosenberg, Biharmonic submanifolds with parallel mean curvature in $\mb{S}^{n} \times \mb{R}$, J. Geom. Anal. 23 (2013), no. 4, 2158–2176. 10.1007/s12220-012-9323-3 MR3107694 1281.58008 D. Fetcu, C. Oniciuc and H. Rosenberg, Biharmonic submanifolds with parallel mean curvature in $\mb{S}^{n} \times \mb{R}$, J. Geom. Anal. 23 (2013), no. 4, 2158–2176. 10.1007/s12220-012-9323-3 MR3107694 1281.58008
D. Fetcu and H. Rosenberg, A note on surfaces with parallel mean curvature, C. R. Math. Acad. Sci. Paris 349 (2011), no. 21-22, 1195–1197. 10.1016/j.crma.2011.10.012 1229.53063 D. Fetcu and H. Rosenberg, A note on surfaces with parallel mean curvature, C. R. Math. Acad. Sci. Paris 349 (2011), no. 21-22, 1195–1197. 10.1016/j.crma.2011.10.012 1229.53063
––––, Surfaces with parallel mean curvature in $\mb{S}^{3} \times \mb{R}$ and $\mb{H}^{3} \times \mb{R}$, Michigan Math. J. 61 (2012), no. 4, 715–729. 10.1307/mmj/1353098510 ––––, Surfaces with parallel mean curvature in $\mb{S}^{3} \times \mb{R}$ and $\mb{H}^{3} \times \mb{R}$, Michigan Math. J. 61 (2012), no. 4, 715–729. 10.1307/mmj/1353098510
––––, On complete submanifolds with parallel mean curvature in product spaces, Rev. Mat. Iberoam. 29 (2013), no. 4, 1283–1306. 10.4171/rmi/757 MR3148604 1294.53055 ––––, On complete submanifolds with parallel mean curvature in product spaces, Rev. Mat. Iberoam. 29 (2013), no. 4, 1283–1306. 10.4171/rmi/757 MR3148604 1294.53055
Y. Fu and A. I. Nistor, Constant angle property and canonical principal directions for surfaces in $\mb{M}^2(c) \times \mb{R}_1$, Mediterr. J. Math. 10 (2013), no. 2, 1035–1049. 10.1007/s00009-012-0219-z MR3045694 1277.53018 Y. Fu and A. I. Nistor, Constant angle property and canonical principal directions for surfaces in $\mb{M}^2(c) \times \mb{R}_1$, Mediterr. J. Math. 10 (2013), no. 2, 1035–1049. 10.1007/s00009-012-0219-z MR3045694 1277.53018
H. Hopf, Differential Geometry in the Large, Lectures Notes in Mathematics 1000, Springer-Verlag, Berlin, 1983. 10.1007/978-3-662-21563-0 0526.53002 H. Hopf, Differential Geometry in the Large, Lectures Notes in Mathematics 1000, Springer-Verlag, Berlin, 1983. 10.1007/978-3-662-21563-0 0526.53002
Z. H. Hou and W.-H. Qiu, Submanifolds with parallel mean curvature vector field in product spaces, Vietnam J. Math. 43 (2015), no. 4, 705–723. 10.1007/s10013-015-0130-6 1329.53085 Z. H. Hou and W.-H. Qiu, Submanifolds with parallel mean curvature vector field in product spaces, Vietnam J. Math. 43 (2015), no. 4, 705–723. 10.1007/s10013-015-0130-6 1329.53085
A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), no. 1, 13–72. 10.1007/bf02564570 0080.15001 A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), no. 1, 13–72. 10.1007/bf02564570 0080.15001
H. F. de Lima and E. A. Lima, Jr., Generalized maximum principles and the unicity of complete spacelike hypersurfaces immersed in a Lorentzian product space, Beitr. Algebra Geom. 55 (2014), no. 1, 59–75. 10.1007/s13366-013-0137-7 1306.53056 H. F. de Lima and E. A. Lima, Jr., Generalized maximum principles and the unicity of complete spacelike hypersurfaces immersed in a Lorentzian product space, Beitr. Algebra Geom. 55 (2014), no. 1, 59–75. 10.1007/s13366-013-0137-7 1306.53056
K. Nomizu and B. Smyth, A formula of Simons' type and hypersurfaces with constant mean curvature, J. Differential Geom. 3 (1969), 367–377. MR266109 0196.25103 10.4310/jdg/1214429059 euclid.jdg/1214429059
K. Nomizu and B. Smyth, A formula of Simons' type and hypersurfaces with constant mean curvature, J. Differential Geom. 3 (1969), 367–377. MR266109 0196.25103 10.4310/jdg/1214429059 euclid.jdg/1214429059
J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), no. 1, 62–105. 10.2307/1970556 0181.49702 J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), no. 1, 62–105. 10.2307/1970556 0181.49702
B. Smyth, Submanifolds of constant mean curvature, Math. Ann. 205 (1973), no. 4, 265–280. 10.1007/bf01362697 MR334102 0262.53037 B. Smyth, Submanifolds of constant mean curvature, Math. Ann. 205 (1973), no. 4, 265–280. 10.1007/bf01362697 MR334102 0262.53037
S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), no. 2, 201–228. 10.1002/cpa.3160280203 0291.31002 S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), no. 2, 201–228. 10.1002/cpa.3160280203 0291.31002
