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Isometric immersions into products of space forms. (English) Zbl 1213.53079

The author proves the existence and uniqueness of an isometric immersion of a Riemannian manifold into the Riemannian product of two spaces forms, extending the result obtained by B. Daniel [Trans. Am. Math. Soc. 361, No. 12, 6255–6282 (2009; Zbl 1213.53075)].

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C40 Global submanifolds
53B25 Local submanifolds

Citations:

Zbl 1213.53075
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References:

[1] Daniel B.: Isometric immersions into $${\(\backslash\)mathbb{S}\^{n}\(\backslash\),\(\backslash\)times\(\backslash\),\(\backslash\)mathbb{R}\(\backslash\), {\(\backslash\)rm and}\(\backslash\), \(\backslash\)mathbb{H}\^{n} \(\backslash\),\(\backslash\)times\(\backslash\), \(\backslash\)mathbb{R}}$$ and applications to minimal surfaces. Trans. Am. Math. Soc. 361, 6255–6282 (2009) · Zbl 1213.53075 · doi:10.1090/S0002-9947-09-04555-3
[2] Dillen F.: Equivalence theorems in affine differential geometry. Geom. Dedicata 32, 81–92 (1989) · Zbl 0684.53012 · doi:10.1007/BF00181438
[3] Dillen F., Nomizu K., Vrancken L.: Conjugate connections and Radon’s theorem in affine differential geometry. Monatsh. Math. 109, 221–235 (1990) · Zbl 0712.53008 · doi:10.1007/BF01297762
[4] Lira, J.H., Tojeiro, R., Vitório, F.: A Bonnet theorem for isometric immersions into product of space forms (preprint) · Zbl 1208.53061
[5] Piccione P., Tausk D.: An existence theorem for G-strucure preserving affine immersions. Indiana Univ. Math. J. 57, 1431–1465 (2008) · Zbl 1163.53009 · doi:10.1512/iumj.2008.57.3281
[6] Yano, K., Kon, M.: Structures on manifolds, series in pure mathematics, 3. World Scientific Publishing Co. · Zbl 0557.53001
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