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user21574
user21574

Consider the homogeneous space $SU(n)/K$, where $K$ is a sub-group of $SU(n)$ and the bi-invariant metric on $SU(n)$.

What is the appropriate quotient metric on the homogeneous space and what are the geodesics? Clearly, for example, $K=U(n-1)$ gives $\mathbb{C}P^n$$\mathbb{C}P^{n-1}$ and the Fubini-Study geodesics which can be expressed in terms of the original geodesics on $SU(n)$. Is any similar formula available for the more general case and is the resulting metric unique (upto constant positive multiple) as in the FS case?

I'm not seeking homogeneous geodesics specifically.

Consider the homogeneous space $SU(n)/K$, where $K$ is a sub-group of $SU(n)$ and the bi-invariant metric on $SU(n)$.

What is the appropriate quotient metric on the homogeneous space and what are the geodesics? Clearly, for example, $K=U(n-1)$ gives $\mathbb{C}P^n$ and the Fubini-Study geodesics which can be expressed in terms of the original geodesics on $SU(n)$. Is any similar formula available for the more general case and is the resulting metric unique (upto constant positive multiple) as in the FS case?

I'm not seeking homogeneous geodesics specifically.

Consider the homogeneous space $SU(n)/K$, where $K$ is a sub-group of $SU(n)$ and the bi-invariant metric on $SU(n)$.

What is the appropriate quotient metric on the homogeneous space and what are the geodesics? Clearly, for example, $K=U(n-1)$ gives $\mathbb{C}P^{n-1}$ and the Fubini-Study geodesics which can be expressed in terms of the original geodesics on $SU(n)$. Is any similar formula available for the more general case and is the resulting metric unique (upto constant positive multiple) as in the FS case?

I'm not seeking homogeneous geodesics specifically.

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Benjamin
Benjamin
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Geodesics on Homogeneous Spaces of $SU(n)$

Consider the homogeneous space $SU(n)/K$, where $K$ is a sub-group of $SU(n)$ and the bi-invariant metric on $SU(n)$.

What is the appropriate quotient metric on the homogeneous space and what are the geodesics? Clearly, for example, $K=U(n-1)$ gives $\mathbb{C}P^n$ and the Fubini-Study geodesics which can be expressed in terms of the original geodesics on $SU(n)$. Is any similar formula available for the more general case and is the resulting metric unique (upto constant positive multiple) as in the FS case?

I'm not seeking homogeneous geodesics specifically.