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• W2090889148 abstract "We obtain a partial description of the totally geodesic submanifolds of a 2-step, simply connected nilpotent Lie group with a left invariant metric. We consider only the case that <italic>N</italic> is nonsingular; that is, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=ad xi colon script upper N right-arrow script upper Z> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>ad</mml:mtext> </mml:mrow> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo>:</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>N</mml:mi> </mml:mrow> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {ad}}xi :mathcal {N} to mathcal {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is surjective for all elements <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=xi element-of script upper N minus script upper Z> <mml:semantics> <mml:mrow> <mml:mi>ξ<!-- ξ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>N</mml:mi> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>xi in mathcal {N} - mathcal {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper N> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>N</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the Lie algebra of <italic>N</italic> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper Z> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>Z</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the center of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper N> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>N</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Among other results we show that if <italic>H</italic> is a totally geodesic submanifold of <italic>N</italic> with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=dimension upper H greater-than-or-equal-to 1 plus dimension script upper Z> <mml:semantics> <mml:mrow> <mml:mi>dim</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>H</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>dim</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>dim H geq 1 + dim mathcal {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <italic>H</italic> is an open subset of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g upper N Superscript asterisk> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>g{N^ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <italic>g</italic> is an element of <italic>H</italic> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N Superscript asterisk> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{N^ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a totally geodesic subgroup of <italic>N</italic>. We find simple and useful criteria that are necessary and sufficient for a subalgebra <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper N Superscript asterisk> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>N</mml:mi> </mml:mrow> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {N}^ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper N> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>N</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be the Lie algebra of a totally geodesic subgroup <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N Superscript asterisk> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{N^ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We define and study the properties of a Gauss map of a totally geodesic submanifold <italic>H</italic> of <italic>N</italic>. We conclude with a characterization of 2-step nilpotent Lie groups <italic>N</italic> of Heisenberg type in terms of the abundance of totally geodesic submanifolds of <italic>N</italic>." @default.
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• W2090889148 title "Geometry of 2-step nilpotent groups with a left invariant metric. II" @default.
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