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Theorems of Gauss-Bonnet and Chern-Lashof types in a simply connected symmetric space of non-positive curvature. (English) Zbl 1048.53041
The Gauss-Bonnet and Chern-Lashof theorems for an \(n\)-dimensional compact immersed submanifold \(M\) in the \(m\)-dimensional Euclidean space \(\mathbb{R}^m\) involving the Euler characteristic \(\chi(M)\) and the \(k\)th Betti numbers \(b_k(M,F)\), are generalized to compact submanifolds in a simply connected symmetric space \(N\) of non-positive curvature. (It is noted that an inequality of Chern-Lashof type is obtained for the case \(N= H^m(-1)\) by E. Teufel in [Colloq. Math. Soc. János Bolyai 46, 1201–1209 (1988; Zbl 0637.53076)]). The proofs are performed by applying the Morse theory to squared distance functions. The corollaries are made for the cases when \(N\) is \(FH^m(c)\), where \(F= \mathbb{R}\), \(\mathbb{C}\), \(\mathbb{H}\) or Cay and \(m= 2\) when \(F= \text{Cay}\). As an example the geodesic sphere in \(FH^m(c)\) is considered.

MSC:
53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C35 Differential geometry of symmetric spaces
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[1] S.S. Chern and R.K. Lashof, On the total curvature of immersed manifolds I, Amer. J. Math., 79 (1957), 306-318. JSTOR: · Zbl 0078.13901
[2] S.S. Chern and R. K. Lashof, On the total curvature of immersed manifolds II, Michigan Math. J., 5 (1958), 5-12. · Zbl 0095.35803
[3] D. Ferus, Totale Absolutkr\(\ddot u\)mmung in Differentialgeometrie und -topologie , Lecture Notes 66 , Springer (1968). · Zbl 0201.23801
[4] S. Helgason, Differential geometry\(,\) Lie groups and symmetric spaces , Academic Press (1978). · Zbl 0451.53038
[5] N. Koike, The Lipschitz-Killing curvature for an equiaffine immersion and theorems of Gauss-Bonnet type and Chern-Lashof type, Results Math., 39 (2001), 230-244. · Zbl 1033.53010
[6] N. Koike, The total absolute curvature of an equiaffine immersion, Results Math., 42 (2002), 81-106. · Zbl 1071.53007
[7] N. Koike, Tubes of nonconstant radius in a symmetric space, Kyushu J. of Math., 56 (2002), 267-291. · Zbl 1027.53053
[8] N.H. Kuiper, Minimal total absolute curvature for immersions, Invent. Math., 10 (1970), 209-238. · Zbl 0195.51102
[9] N.H. Kuiper, Tight embeddings and maps. Submanifolds of geometrical class three in \(E^n\), The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979) Springer (1980), 97-145. · Zbl 0461.53033
[10] J. Milnor, Morse theory , Ann. Math. Stud. 51 (1963), Princeton University Press. · Zbl 0108.10401
[11] K. Nomizu and L. Rodriguez, Umbilical submanifolds and Morse functions, Nagoya Math. J., 48 (1972), 197-201. · Zbl 0246.53050
[12] C.L. Terng and G. Thorbergsson, Submanifold geometry in symmetric spaces, J. Differential Geom., 42 (1995), 665-718. · Zbl 0845.53040
[13] E. Teufel, Differential topology and the computation of total absolute curvature, Math. Ann., 258 (1982), 471-480. · Zbl 0464.53049
[14] E. Teufel, On the total absolute curvature of immersions into hyperbolic spaces, Topic in Differential Geometry Vol II , North-Holland (1988), 1201-1210. · Zbl 0637.53076
[15] T.J. Willmore and B.A. Saleemi, The total absolute curvature of immersed manifolds, J. London Math. Soc., 41 (1966), 153-160. · Zbl 0136.42902
[16] T.J. Willmore, Total curvature in Riemannian geometry , Ellis-Horwood (1982). · Zbl 0501.53038
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